Bring a large pot of water to a boil and cook the shucked corn for 3 minutes.
Transfer corn to a large bowl of ice water to stop cooking, then when cool, cut the kernels from the cobs.
Place the corn kernels, sliced zucchini, and chopped thyme together in a grilling basket and set aside.
Brush or rub 1 tablespoon olive oil over the chicken breasts, then combine the cumin, 1/2 teaspoon kosher salt, and 1/4 teaspoon cracked pepper and rub the mixture evenly onto both sides of the chicken.
Preheat the grill to medium-high and lightly spray or oil the grates.
Grill the chicken 4–5 minutes per side until cooked through and no longer pink in the center, then remove and set aside.
Place the vegetable grilling basket over the grill and cook 5–8 minutes, tossing occasionally, until zucchini is tender and corn is slightly charred.
Arrange the grilled chicken on a platter, top with the grilled succotash, then finish with the thinly sliced red onions, chopped green onions, and crumbled Cojeta cheese. Season with additional salt and pepper if desired and serve immediately.