Melt the butter in a medium saucepan over medium-low heat. Add the onion and ½ teaspoon of salt and sauté until the onion is golden brown (about 3 minutes). Add in the garlic and cook until fragrant (about 30 seconds).
Stir in the crushed tomatoes and bring to a boil. Reduce heat and simmer for 10 minutes. Remove the pan from the heat and stir in Parmesan cheese, fresh basil, and olive oil.
Season with salt and pepper to taste.
Season the chicken breasts on both sides with salt and pepper. Lightly grease the grates of an outdoor grill. Heat the grill to medium-high heat. Add the chicken breasts and cook until browned on one side (3-5 minutes). Flip the chicken breast and cook on the other side.
Cook the chicken until it is browned on both sides and fully cooked (the internal temperature has reached 165 Degrees F.). During the last few minutes of cooking, top each chicken breast with about 2 tablespoons of the tomato sauce, and some of the mozzarella cheese.