Season chicken breasts with salt and pepper. If using large breasts, slice in half horizontally for quicker cooking.
In a large skillet over medium heat, heat olive oil and butter until melted. Add minced garlic and sliced sweet onions. Sauté for about 5 minutes until onions are softened and slightly caramelized.
Add seasoned chicken breasts to the skillet. Cook about 4-5 minutes on each side until golden brown.
Pour in diced tomatoes with their juices and balsamic vinegar. Sprinkle granulated sugar if using to balance acidity.
Sprinkle dried oregano, basil, rosemary, and thyme over chicken and tomatoes. Stir gently to combine and coat chicken well.
Bring mixture to a gentle simmer, reduce heat to low, cover, and cook for about 15-20 minutes or until chicken reaches 165°F (75°C) internal temperature.
Remove from heat and serve chicken hot with sauce and vegetables spooned over. Pairs well with rice, pasta, or fresh salad.