This is a phase I, open-label, non-randomized study that will enroll pediatric and young adult research participants with relapsed or refractory non-CNS solid tumors to evaluate the safety, feasibility, and efficacy of administering T cell products derived from the research participant's blood that have been genetically modified to express a EGFR-specific receptor (chimeric antigen receptor, or CAR) that will target and kill solid tumors that express EGFR and the selection-suicide marker EGFRt. EGFRt is a protein incorporated into the cell with our EGFR receptor which is used to identify the modified T cells and can be used as a tag that allows for elimination of the modified T cells if needed. On Arm A of the study, research participants will receive EGFR-specific CAR T cells only. On Arm B of the study, research participants will receive CAR T cells directed at EGFR and CD19, a marker on the surface of B lymphocytes, following the hypothesis that CD19+ B cells serving in their normal role as antigen presenting cells to T cells will promote the expansion and persistence of the CAR T cells. The CD19 receptor harbors a different selection-suicide marker, HERtG. The primary objectives of the study will be to determine the feasibility of manufacturing the cell products, the safety of the T cell product infusion, to determine the maximum tolerated dose of the CAR T cells products, to describe the full toxicity profile of each product, and determine the persistence of the modified cell in the subject's body on each arm. Subjects will receive a single dose of T cells comprised of two different subtypes of T cells (CD4 and CD8 T cells) felt to benefit one another once administered to the research participants for improved potential therapeutic effect. The secondary objectives of this protocol are to study the number of modified cells in the patients and the duration they continue to be at detectable levels. The investigators will also quantitate anti-tumor efficacy on each arm. Subjects who experience significant and potentially life-threatening toxicities (other than clinically manageable toxicities related to T cells working, called cytokine release syndrome) will receive infusions of cetuximab (an antibody commercially available that targets EGFRt) or trastuzumab (an antibody commercially available that targets HER2tG) to assess the ability of the EGFRt on the T cells to be an effective suicide mechanism for the elimination of the transferred T cell products.

Inclusion Criteria:

• First 2 subjects enrolled and treated in both Arm A and Arm B: age ≥ 15 and ≤ 30 years
• Subsequent subjects: age ≥ 1 and ≤30years
• Histologically diagnosed malignant, non-CNS solid tumor expressing EGFR
• Evidence of refractory or recurrent disease
• Able to tolerate apheresis or has apheresis product available for use in manufacturing
• Life expectancy ≥ 8 weeks
• Lansky or Karnofsky score ≥ 50
• Recovered from significant acute toxic effects of all prior chemotherapy, immunotherapy, and radiotherapy
• If no apheresis product or T cell product is available,≥ 7 days post last chemotherapy/biologic therapy administration
• If no apheresis product or T cell product is available,≥ 3 half lives or 30 days, whichever is shorter, post last dose of anti-tumor antibody therapy (including check point inhibitor)
• Prior genetically modified cell therapy is allowed if not detectable at enrollment.
• If no apheresis product or T cell product is available,≥ 6 weeks post last dose of myeloablative therapy and allogeneic or autologous stem cell transplant
• Subjects who receive autologous stem cell infusion following non-myeloablative therapy are eligible once all other eligibility requirements are met
• If no apheresis product or T cell product is available,≥ 7 days post last systemic corticosteroid therapy (physiologic replacement dosing is allowed)
• If no apheresis product or T cell product is available, subjects with neuroblastoma must be ≥ 12 weeks from I131 MIBG therapy.
• Adequate organ function
• Adequate laboratory values
• Patients of childbearing potential must agree to use highly effective contraception

Exclusion Criteria:

• Presence of active malignancy other than primary malignant solid tumor diagnosis
• Current relevant CNS pathology
• Presence of active GVHD, or receiving immunosuppressive therapy for treatment or prevention of GVHD within 4 weeks prior to enrollment
• Presence of active severe infection
• Presence of primary immunodeficiency syndrome
• Receiving external beam radiation therapy at time of enrollment
• Receiving any anti-cancer agents or chemotherapy
• Pregnant or breastfeeding
• Unwilling to provide consent/assent for participation in the study and 15 year follow up period
• Presence of any condition that, in the opinion of the investigator, would prohibit the patient from undergoing treatment under this protocol