Preheat the oven to broil and lightly oil a baking sheet.
Set up three shallow bowls: one with 1/4 cup flour, one with the beaten egg, and one with 1/2 cup panko mixed with 1/4 cup grated Parmesan.
Season both sides of the chicken breasts with salt and pepper, then dredge each piece in flour, dip in egg, and coat evenly with the panko-Parmesan mixture.
Heat 3 teaspoons of the olive oil in a large skillet over medium heat. Working in batches if needed, cook the breaded chicken 2 minutes per side until golden.
Transfer the seared chicken to the prepared baking sheet, top each piece with sliced mozzarella, and broil in the oven about 1 minute until the cheese melts.
In the same skillet, add the remaining 1 teaspoon olive oil and heat over medium. Add the zucchini and summer squash and cook 1 minute, stirring.
Add the cherry tomatoes and crushed garlic to the skillet and cook about 3 minutes, until the vegetables are crisp-tender.
Serve the sautéed vegetables spooned over the broiled chicken and sprinkle with chopped fresh basil.