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In a large oven-proof skillet over medium-high heat, heat oil. Season both sides of chicken breasts with salt, pepper, and Italian seasoning and add to skillet. Cook until cooked through and golden on both sides, about 8 minutes per side. Remove chicken and set aside. In the same skillet over medium heat, melt butter. Add garlic and cook, stirring with a wooden spoon constantly, until the garlic is fragrant, about 1 minute. Add the white wine to deglaze the pan. Using the wooden spoon, scrape the bottom of the skillet to release any leftover chicken bits. (They add a lot of flavor!) Cook until the wine has reduced by about half. Add the chicken broth, asparagus, cherry tomatoes, squash and thyme. Season with salt and pepper and cook until the vegetables are tender. Stir in heavy cream, then add chicken back to the skillet. Preheat broiler. Top each chicken breast with a slice of mozzarella then transfer the whole skillet to the oven. Broil until the cheese is bubbly, about 2 minutes. Garnish with parsley and serve warm over rice.