Slice each pounded chicken breast in half, making 2 portions each. You should have 8 total chicken cutlets. You can also use 8 boneless chicken thighs.
In a large bowl, combine chicken cutlets with olive oil and barbecue seasoning (see 'Tips' below for recipe for homemade seasoning). Toss well to coat chicken in seasoning. Cover bowl with plastic wrap and refrigerate until ready to cook, up to 1 day.
When ready to cook, preheat gas grill to medium-high heat. Drizzle zucchini and squash with a little olive oil. Place chicken, zucchini and squash on hot grates. Cook 4-5 minutes per side, until chicken juices run clear.
Remove chicken and veggies from grill. Place on a platter, and top each chicken piece with 1 slice mozzarella and 1 slice tomato. Sprinkle with fresh basil. Drizzle with olive oil and balsamic vinegar.
Make your own 1/4 cup barbecue seasoning with 4 teaspoons chili powder, 2 teaspoons garlic powder, 2 teaspoons onion powder, 1 teaspoon ground cumin, 1 teaspoon dried oregano, 1 teaspoon kosher salt, and 1/2 teaspoon black pepper.
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