Evaluating the Safety and Tolerability of Baricitinib in Patients With Job Syndrome With Lupus-Like Disease and/or Atopic Dermatitis

* INCLUSION CRITERIA: To be eligible to participate in this study, an individual must meet all of the following criteria: 1. Must be able to understand and provide informed consent or assent. 2. Aged \>=12 years. 3. Documented STAT3 variant causing hyper-IgE syndrome. 4. Enrollment in NIH protocol 00-I-0159, Natural History, Management, and Genetics of the Hyperimmunoglobulin E Recurrent Infection Syndrome (HIES). a. Presence of SLE and/or AD as follows: SLE patients should meet either Systemic Lupus International Collaborating Clinics (SLICC) or 2019 American College of Rheumatology (ACR)/European League Against Rheumatism (EULAR) SLE classification criteria. AD is defined as EASI tool score \>=16 and body surface area tool score of 10% at screening. 5. Ability to take oral medication and be willing to adhere to the study intervention regimen. 6. For individuals on glucocorticoids, the dose must be less than 10 mg daily and stable for the 30 days prior to Day 0. 7. For individuals on hydroxychloroquine or other antimalarials such as chloroquine or quinacrine, the dose must have been stable for 90 days prior to Day 0. The maximum allowed dose is hydroxychloroquine 400 mg/day or 6.5 mg/kg/day, whichever is greater. The maximum allowed dose for chloroquine phosphate is 500 mg daily, and for quinacrine is 100 mg daily. 8. Individuals may be on lipid-lowering medications if initiated at least 90 days prior to Day 0, and the dose must be stable for 30 days prior to Day 0. 9. Individuals of reproductive potential must agree to use at least one highly effective method of contraception when engaging in sexual activities that can result in pregnancy while on study drug. Acceptable methods of contraception include: * Intrauterine device (IUD) * Bilateral tubal ligation * Abstinence * Vasectomized partner * Hormonal contraception used in combination with barrier method: progestogen containing (oral, intravaginal, transdermal) or progestogen-only (oral, injectable, implantable) starting 30 days prior to initiation of baricitinib EXCLUSION CRITERIA: An individual who meets any of the following criteria will be excluded from participation in this study: 1. Known history of hypersensitivity to baricitinib or other JAK inhibitors. 2. Current or recent use of any investigational drug/intervention (within 6 months or 5 half-lives, whichever is longer, prior to Day 0) except for COVID-19 vaccines or therapies that have been granted an FDA emergency authorization. 3. Scheduled to participate in another clinical study involving an investigational drug during the course of this study. 4. Use of systemic immunosuppressive or immune-modulating agents within 90 days prior to Day 0, except systemic steroids \<=10 mg of prednisone equivalent per day. 5. Current or prior treatment with rituximab in the 6 months prior to Day 0. 6. Current treatment with methotrexate, mycophenolate mofetil, other less common immunomodulatory drugs such as those falling into the class of disease-modifying antirheumatic drugs (DMARDs), belimumab, and other immunosuppressive biologics not otherwise specified herein. Participants previously on methotrexate, mycophenolate mofetil, azathioprine, tacrolimus, cyclosporine, or belimumab, other immunosuppressive biologics, or DMARDs should have been withdrawn from the drug for at least 90 days prior to Day 0. 7. Treatment with cyclophosphamide and pulse methylprednisolone within 6 months prior to Day 0. 8. Hypercholesterolemia: Values after 8- to 12-hour fasting blood specimen: total cholesterol \>250 mg/dL or LDL \>180 mg/dL or hypertriglyceridemia (triglyceride \>300 mg/dL) within 90 days prior to Day 0. 9. History of alcohol or drug abuse within 6 months prior to Day 0. 10. Presence of 1 or more of the following clinically significant laboratory abnormalities: 1. Serum ALT \>=3 times ULN. 2. Serum total bilirubin \>=2 times ULN. 3. ANC \<=750 cells/microL. 4. Hemoglobin \<=9.0 g/dL. 5. Platelet count \<=100,000/microL. 6. Serum creatinine \>=2 times ULN. 11. Planned or anticipated major surgical procedure during the study. 12. Plans to receive any live vaccines within 1 month of the anticipated first dose of baricitinib. 13. Known or suspected immune-dysregulatory disorders besides Job s syndrome, lupus-like disease, and/or AD. 14. Active invasive opportunistic infections (eg, non-TB mycobacterial infections, histoplasmosis, listeriosis, coccidioidomycosis, pneumocystis pneumonia, aspergillosis) despite infection resolution or otherwise recurrent infections of abnormal frequency or prolonged infections suggesting an immune-compromised status as judged by the investigator. 15. Known active TB. Participants with treated LTB will be eligible to participate. Participants with untreated LTB will not be excluded but will be evaluated by an infectious disease consultant and may become eligible for trial based on infectious disease consultant recommendations. 16. Infection with HIV. 17. Untreated infection with hepatitis B or C. 18. Unwillingness to receive prophylactic entecavir (or similar), only for individuals with evidence of clearance of hepatitis B with positive hepatitis B core and surface antibody and negative hepatitis B surface antigen and PCR. 19. BK or JC viremia at screening visit. 20. Active infection that requires the use of oral or intravenous antimicrobials that remains unresolved at least 14 days prior to the administration of the first dose of study medication. 21. Individuals with active renal or central nervous system disease or a high activity level in any organ system (except articular) that requires immediate immunosuppressive therapy as determined by the investigator. 22. History of cancer, with the exceptions of basal cell carcinoma, localized squamous cell carcinoma of the skin, or in situ carcinoma of the cervix, provided the participant is in remission and curative therapy was completed at least 12 months prior to screening. History of other malignancies are also permitted provided that the individual is in remission and curative therapy was completed at least 5 years prior to screening. 23. Planned or anticipated use of any prohibited medications and procedures during the study. 24. Pregnancy or current breastfeeding. 25. Currently receiving hemodialysis or peritoneal dialysis. 26. Past or current medical problems or findings from physical examination, electrocardiogram, or laboratory testing that are not listed above, which, in the opinion of the investigator, may pose additional risk from participation in the study, may interfere with the individual s ability to comply with study requirements or that may impact the quality or interpretation of the data obtained from the study. These may include, but are not limited to: 1. Known coronary artery aneurysm. 2. Known history of arterial or venous thrombosis or at high risk for clotting disorder. 3. Known history of PE or DVT in the past. 4. Psychiatric illness or history of medical non-compliance that the study team feels will make the individual unlikely to complete the study. 5. Significant impairment of major organ function (lung, heart, liver, kidney) or any condition that, in the opinion of the investigator, would jeopardize the individual s safety following exposure to the study drug. 27. Individuals with known increased risk factors for MACE including a history of: 1. Ischemic heart disease (eg, history of acute myocardial infarction). 2. Heart failure. 3. Cardiomyopathy. 4. Severe valvular heart disease. 5. Significant arrhythmias. 6. Chronic renal failure. 7. Cerebrovascular accident or transient ischemic attack. 8. Uncontrolled diabetes mellitus. 9. Uncontrolled hypertension. 10. Current smokers or former smokers with less than 3 years since complete cessation and/or \>20 pack-years of smoking history. 28. History of idiopathic GI perforation or diverticulitis with high risk of perforation. 29. Treatment with strong organic anion transporter 3 inhibitors (OAT3) (eg, probenecid) due to drug interactions. 30. Uncontrolled thyroid disease as per principal investigator or medically responsible investigator.