Study of Miransertib (MK-7075) in Participants With PIK3CA-related Overgrowth Spectrum and Proteus Syndrome (MOSAIC) (MK-7075-002)

Inclusion Criteria: Part A * Signed informed consent and, when applicable, signed assent * Male or female participants ≥ 2 years old with body surface area (BSA) of ≥ 0.33 m\^2 * Have a clinical diagnosis of PROS or PS with documented somatic PIK3CA or serine-threonine protein kinase (AKT1) mutations * Archival or fresh overgrowth tissue sample available to be shipped to Sponsor or designee * Have poor prognosis, significant morbidity, and/or progressive disease (e.g., worsening of the disease/increase in number or size of the overgrowth lesions in the last 12 months) * Have measurable disease (at least one overgrowth lesion that can be accurately measured in size by imaging and/or linear or circumference measure) * Adequate organ function based on screening laboratory values * If a female is of child-bearing potential, documentation of a negative pregnancy test is required prior to enrollment. Sexually active participants (male and female) must agree to use double-barrier contraceptive measures, oral contraception, or avoidance of intercourse while on study and for up to 90 days after ending treatment * Ability to complete the Quality of Life (QoL) questionnaires by the participant or his/her caregiver Part B: * Signed consent form and when applicable, signed assent * Archival or fresh overgrowth tissue sample available to be shipped to Sponsor or designee * Except for Cohort 4, clinically progressive or worsening disease defined as an increase in number or size of the overgrowth lesion(s) in the last 6 months as assessed by the Investigator * Adequate organ function based on screening laboratory values * Male or female participants of child-producing potential must agree to use double-barrier contraceptive measures, oral contraception, or avoidance of intercourse during the study and for 90 days after the last dose of miransertib * Ability to complete the study questionnaires by the participant or his/her caregiver Cohort 1 (PROS) specific criteria * Male or female participants ≥ 2 years and ≤30 years of age with BSA of ≥ 0.33 m2 * Have clinical diagnosis of PROS per Diagnostic Criteria for PROS and documented somatic PIK3CA variant * Have at least one lesion that can be measured by study- standardized volumetric MRI (eligibility to be confirmed by blinded independent central imaging review \- Cohort 2 (PS) specific criteria * Male or female participants ≥ 2 years and ≤18 years of age with BSA of ≥ 0.33 m2 * Have clinical diagnosis of PS per Diagnostic Criteria for PS and documented somatic AKT1 variant * Have at least one plantar cerebriform connective tissue nevus (CCTN) and pre-CCTN lesion that can be measured by standardized photography * Cohort 3 specific criteria: Male or female participants ≥2 years old with BSA of ≥ 0.33 m2 and who fail to meet the eligibility criteria for Cohorts 1 or 2 * Cohort 4 (PROS or PS) specific criteria: participants previously treated with miransertib or currently receiving miransertib under Compassionate Use/Expanded Access. Participants should meet the age criterion by/on the date of the first dose, Cycle 1 Day 1 Exclusion Criteria Part A: * History of Type 1 or 2 uncontrolled diabetes mellitus requiring regular medication (other than metformin or other oral hypoglycemic agents) or fasting glucose ≥ 160 mg/dL (if \> 12 years old) and ≥ 180 mg/dL (if ≤ 12 years old) at the screening visit * History of significant cardiac disorders: * Myocardial infarction (MI) or congestive heart failure defined as Class II-IV per the New York Heart Association (NYHA) classification within 6 months of the first dose of miransertib (MI occurring \> 6 months of the first dose of miransertib will be permitted) * Grade 2 (per NCI CTCAE version 4.03) or worse conduction defect (e.g., right or left bundle branch block); left ventricular ejection fraction (LVEF) \< 50% assessed by echocardiogram/multigated acquisition (MUGA) scan * Major surgery, radiotherapy, or immunotherapy within four weeks of the first dose of miransertib * Any experimental systemic therapy for the purpose of treating PROS or PS (e.g., sirolimus, everolimus, high dose steroids) within two weeks of the first dose of miransertib, except for participants who were previously or are currently treated with miransertib under a Compassionate Use/Expanded Access program * Intolerance of or severe toxicity attributed to v-Akt murine thymoma viral oncogene homolog (AKT) inhibitors (e.g., miransertib, uprosertib, afuresertib, ipatasertib) * Concurrent severe uncontrolled illness not related to PROS or PS (ongoing or active infection, known human immunodeficiency virus (HIV) infection, malabsorption syndrome, psychiatric illness/substance abuse/social situation that would limit compliance with study requirements) * Pregnant or breastfeeding * Inability to comply with study evaluations or to follow drug administration guidelines Part B * History of Type 1 diabetes mellitus or Type 2 uncontrolled diabetes mellitus requiring regular medication (other than metformin or other oral hypoglycemic agents) or fasting glucose ≥ 160 mg/dL (if \> 12 years old) and ≥ 180 mg/dL (if ≤ 12 years old) at the screening visit * History of significant cardiac disorders: * Myocardial infarction (MI) or congestive heart failure defined as Class II-IV per the New York Heart Association (NYHA) classification within 6 months of the first dose of miransertib (MI occurring \> 6 months of the first dose of miransertib will be permitted) * Grade 2 (per current version of National Cancer Institute \[NCI\] Common Terminology Criteria for Adverse Events \[CTCAE\]) or worse conduction defect (e.g., right or left bundle branch block) * Major surgery or locoregional therapy within four weeks of the first dose of miransertib * Any experimental systemic therapy for the purposes of treating PROS or PS (e.g., sirolimus, everolimus, high dose steroids) within two weeks of the first dose of miransertib * Intolerance of or severe toxicity attributed to AKT inhibitors (e.g., miransertib, uprosertib, afuresertib, ipatasertib) * Concurrent severe uncontrolled illness not related to PROS or PS (e.g. ongoing or active infection, known HIV infection, malabsorption syndrome, psychiatric illness/substance abuse/social situation that would limit compliance with study requirements) * Pregnant or breastfeeding * Inability to comply with study evaluations or to follow drug administration guidelines