Preheat the oven to 350°F (175°C).
Rinse the chicken breasts and pat them dry with paper towels.
Season both sides of the chicken with half of the salt and half of the pepper (about 1/4 teaspoon each).
Beat the eggs in a shallow bowl. Dip each chicken breast into the beaten eggs, letting excess drip off.
Spread the grated Parmesan on a plate and press each egg-coated breast into the cheese to coat both sides.
In a large bowl, toss the zucchini, yellow squash, sliced garlic, halved cherry tomatoes, thyme, the remaining salt and pepper, and the olive oil until evenly coated.
Spread the vegetable mixture in an even layer in a 13 x 9-inch baking dish. Arrange the Parmesan-coated chicken breasts on top of the vegetables.
Bake for 35 to 40 minutes, until the chicken reaches an internal temperature of 165°F (74°C) and the vegetables are tender.
Remove from the oven and let rest a few minutes before serving the chicken with the roasted vegetables.