Place the oven rack in the center and preheat the oven to 400℉ (200℃).
Slice the zucchini, yellow squash, red onion, and tomato into ¼-inch slices and set them aside.
Place the chicken breasts on a cutting board and pat them dry with a paper towel. Make slits in the chicken breasts, cutting about three-quarters of the way through, approximately every ½ inch. Avoid cutting into the ends of the chicken.
Lightly grease an oven-safe baking dish with about 1 tablespoon of olive oil. Arrange the chicken breasts in the dish and stuff each slit with alternating slices of the prepared vegetables for a colorful presentation.
Drizzle about 1 tablespoon of olive oil over the chicken and sprinkle with Italian seasoning, salt, and pepper.
Place the dish in the oven and bake uncovered for a total of 25 minutes.
Halfway through baking, carefully remove the dish from the oven. Top the chicken with shredded mozzarella cheese, then return the dish to the oven to finish baking.
When the timer goes off, use a meat thermometer to check the chicken's internal temperature. It should reach at least 165℉ (75℃) in the thickest part of the meat.
Once the chicken is fully cooked, remove it from the oven and let it rest for about 5 minutes before serving. Garnish with chopped fresh parsley if desired.