Combine chicken, olive oil, garlic, red pepper flakes, salt, and black pepper in a large bowl; toss to coat.

Heat a large pot or Dutch oven over medium-high heat. Add chicken mixture and cook, stirring frequently, until chicken begins to brown, about 5 minutes. Add yellow bell pepper, onion, zucchini, and balsamic vinegar and cook until onion begins to soften, about 5 minutes.

Stir mushrooms and 1 tablespoon basil into chicken mixture and cook until mushrooms have softened, about 5 minutes. Add mozzarella cheese and pesto sauce; toss to coat. Stir in cherry tomatoes and cook until warmed through, 2 to 3 minutes.

Spoon chicken mixture into toasted buns and top each sandwich with 1/4 cup feta cheese and 1 1/2 teaspoons basil.