Season chicken with salt and pepper. Heat 1 tbsp olive oil in a large skillet over medium-high heat and cook chicken 4–6 minutes until browned and cooked through; remove.
Add remaining oil and garlic to skillet; sauté 30 seconds. Stir in tomatoes and cook 2–3 minutes until they start to soften.
Pour in broth and balsamic, add pasta, cover and simmer per pasta package until al dente, stirring occasionally.
Return chicken to pan, stir in mozzarella and basil off heat so cheese softens but doesn’t fully melt. Adjust seasoning and sprinkle red pepper flakes if using.