Preheat the oven to 350°F (175°C).
Season the chicken on both sides with 1 tablespoon of oil, basil, oregano, salt, and pepper.
Brown the chicken: Heat a large oven-proof skillet over medium-high heat. Add the remaining oil, then the chicken. Sear the chicken quickly, flipping once, until golden on both sides. Remove to a plate.
Prepare the glaze: Mix the diced red onion, garlic, balsamic vinegar, and brown sugar. Pour the mixture into the skillet and simmer, stirring, for 3 minutes.
Bake: Return the chicken to the pan and spoon the sauce over it. Place the pan in the oven and roast until the chicken is cooked through (165°F or 74°C), about 10-20 minutes depending on the thickness of the breasts. Remove from the oven.
Turn the oven to broil.
Top each chicken breast with a slice of fresh mozzarella cheese. Broil just until the cheese has melted.
Serve immediately, topped with fresh sliced tomatoes and fresh-cut basil. Spoon the pan sauce over the top.
Make it a meal by serving the Chicken Caprese with a green salad and fresh artisan bread.