In a small bowl, whisk together the ghee, vinegar, sugar, garlic, salt, and pepper. Butterfly the chicken (or cut in half) and place in a shallow dish. Pour about half of the vinegar marinade over it; cover and set aside the other half. Cover the dish with the chicken and allow to marinate in the refrigerator for about 4 hours.
Preheat a skillet or the grill. Grease with butter. Grill the chicken just until cooked through.
Place the cooked breasts on a greased baking sheet. Spoon the vinegar mixture evenly over each breast and top evenly with the diced tomatoes and chopped basil.
Sprinkle with mozzarella.
Put the pan in a 350° Fahrenheit oven and let bake just until the cheese has melted, about 10 minutes.
Allow to rest for 6 to 10 minutes before serving.