In a large skillet over medium heat, heat oil. Add chicken and season with salt, pepper, and half the Italian seasoning. Cook 8 minutes, then flip and season other side. Cook 8 minutes more. Transfer to a plate to let rest.
Add garlic to skillet and cook until fragrant, 1 minute, then add balsamic vinegar to deglaze pan, scraping up brown bits with a wooden spoon. Add crushed tomatoes, basil, and crushed red pepper flakes and stir until combined and warmed through, 3 minutes.
Return chicken to pan and nestle in sauce. Top each breast with 1 slice mozzarella and cover with a lid until melted.