Cook chicken: Heat olive oil over medium high heat. Season chicken with salt and pepper, then cook until golden-brown and cooked through, about 6 minutes per side. Set aside on a plate and tent with foil.
Make sauce: Add chicken broth to pan, scraping any browned bits off the bottom. Stir in balsamic vinegar, brown sugar (if using), Dijon mustard, garlic and Italian seasoning. Simmer over medium heat until thickened to your preferred consistency. Check for seasoning and add more salt and pepper if you like.
Cook tomatoes: Add tomatoes to the pan and simmer for 3-4 minutes, or until softened. Do not overcook or they'll fall apart.
Finish dish: Add chicken back to the pan and simmer for 2-3 minutes or until chicken is heated through, spooning balsamic sauce over the chicken as it cooks. Remove pan from heat and top each chicken breast with one or two slices of mozzarella cheese. Close with a lid until cheese has melted, a few minutes. Serve with sliced basil.