Combine 1 tablespoon olive oil, zucchini, summer squash, and tomatoes in a bowl; toss to combine. Season with 1/2 teaspoon salt and 1 1/2 teaspoons cantanzaro herbs; mix well. Season chicken breasts with the remaining cantanzaro herbs, salt, and pepper on all sides.
Heat remaining tablespoon olive oil in a large skillet over medium-high heat and cook chicken until browned on each side, 3 to 4 minutes per side. Remove chicken from skillet and set aside. Add seasoned vegetables to skillet and cook until tomatoes begin to break down, 4 to 5 minutes. Stir in garlic and cook until fragrant, about 1 minute.
Stir cream cheese and milk into the tomato mixture in the skillet until well combined. Add Parmesan cheese and mix until well incorporated into the sauce. Return cooked chicken to the skillet and season with salt and pepper. Heat until warmed through.
Serve each chicken breast topped with vegetable ragout and garnish with shaved Parmesan cheese if desired.