CARv3-TEAM-E T Cells in Glioblastoma

Inclusion Criteria: * Safety Run In Arm and ARM 1: Recurrent GBM, EGFRvIII mutant * Participants must have histologically confirmed recurrent GBM or molecular features of GBM with presence of EGFRvIII mutation detected at initial diagnosis. MGMT methylated, unmethylated, or unknown is allowed. * Participants must be at first progression or recurrence and plan is for biopsy or surgical debulking. Participants must have at least received prior radiation. Prior temozolomide is not required if the participant is MGMT unmethylated. Participants must be 2 weeks from prior alkylating therapy or immunotherapy and ≥ 5 half-lives from another investigational agent before proceeding with collection or treatment. No washout is required from radiation since participants will need histological confirmation of recurrence to participate. * ARM 2: Newly Diagnosed GBM, EGFRvIII mutant (will only open once safety is confirmed in Arms 1 and 3) * Participants must have histologically confirmed newly diagnosed GBM with presence of EGFRvIII mutation and their tumors must be MGMT unmethylated. * Treatment planned with involved field radiation alone without concomitant or sequential temozolomide. * ARM 3: Recurrent GBM, EGFRvIII negative * Participants must have histologically confirmed recurrent GBM with EGFR amplification but no EGFRvIII mutation based on initial diagnostic tissue. * Participants must be at first recurrence and plan is for biopsy or surgical debulking. Participants must have at least received prior radiation. Prior temozolomide is not required if the participant is MGMT unmethylated. Participants must be 2 weeks from prior alkylating therapy or immunotherapy and ≥ 5 half-lives from another investigational agent. No washout is required from radiation since participants will need histological confirmation of recurrence to participate. * ARM 1: Recurrent GBM, EGFRvIII mutant and ARM 3: Recurrent GBM, EGFRvIII negative: * Must be at least 3 months from completion of radiation or evidence of progression is outside the high dose radiation field. * Safety Run-In Arm and ARM 1: Recurrent GBM, EGFRvIII mutant and ARM 3: Recurrent GBM, EGFRvIII negative: * Participants must have measurable disease, defined as at least one lesion ≥10 mm (≥1 cm) with MRI. See Section 11 (Measurement of Effect) for the evaluation of measurable disease. * ALL ARMS: * Patients cannot have posterior fossa or intramedullary spine-only disease. Leptomeningeal disease is allowed anywhere in the neuroaxis. See Section 11 (Measurement of Effect) for the evaluation of measurable disease. * Resolution of AEs from any prior systemic anticancer therapy or radiotherapy to Grade 1 or baseline (except Grade 2 alopecia and Grade 2 sensory neuropathy) * Medically able and willing to undergo placement of an Ommaya reservoir. * Steroid dose anticipated to be ≤ 4 mg of dexamethasone a day or equivalent at time of first CAR-v3-TEAM-E infusion. * Age ≥18 years * Karnofsky ≥60% * Must be able to undergo an MRI with contrast. * Life expectancy of greater than 3 months. * Participants must have adequate organ and marrow function as defined below: * Absolute neutrophil count ≥1,000/mcL * Platelets ≥80,000/mcL * Total bilirubin ≤ institutional upper limit of normal (ULN); For patients with Gilbert's syndrome, total bilirubin can be ≤ 3xULN. * AST(SGOT)/ALT(SGPT) ≤3 × institutional ULN * CrCl ≥ 60 mL/min * Participant has no prior history of malignancy, unless the subject has been free of the disease for ≥5 years with the exception of the following noninvasive malignancies: * Basal cell carcinoma of the skin * Squamous cell carcinoma of the skin * Carcinoma in situ of the cervix * Carcinoma in situ of the breast * Incidental histologic finding of prostate cancer (T1a or T1b) or prostate cancer that is curative * Left ventricular ejection fraction \>50% as determined by TTE. * The effects of CARv3-TEAM-E on the developing human fetus are unknown. For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of CARv3-TEAM-E administration. * Ability to understand and the willingness to sign a written informed consent document. Exclusion Criteria: * Intraparenchymal posterior fossa disease * Intramedullary spinal disease as the only site of disease. * Prior EGFRvIII targeted therapies. * Prior bevacizumab treatment. * Treatment with an any prior gene-therapy or gene-modified cellular therapy. * Patients with a VP shunt or patients needing a shunt in the immediate future are excluded from participating * Ongoing treatment with chronic immunosuppressants (e.g., cyclosporine or systemic steroids above physiologic dosing). Intermittent topical, inhaled, or intranasal corticosteroids are allowed * Participants who have not recovered from adverse events due to prior anti-cancer therapy (i.e., have residual toxicities \> Grade 1) with the exception of alopecia. * Participants who are receiving any other investigational agents. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to CARv3-TEAM-E (ex. cetuximab). * Participants with uncontrolled intercurrent illness. * Human immunodeficiency virus (HIV)-infected participants are not eligible. * Participants with evidence of chronic hepatitis B virus (HBV) infection or active hepatitis C virus (HCV) infection are not eligible. * Participants with psychiatric illness/social situations that would limit compliance with study requirements. * Pregnant women are excluded from this study because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with CARv3-TEAM-E , breastfeeding should be discontinued if the mother is treated with CARv3-TEAM-E. * For Arm 2, prior to CARv3-TEAM-E Infusion, the following criteria should be confirmed in addition to the relevant criteria above: * Participants must have completed 75% of the planned 6 weeks of involved field radiation without temozolomide * Tumor location and size criteria as in 3.1.7 above. * Prior cancer directed therapy other than radiation is not allowed.