Heat 1 teaspoon oil in a large nonstick skillet over medium heat. Add onion; sauté 2 minutes. Add mushrooms; sauté 3 minutes. Stir in spinach; remove from heat. Stir in cheese, Italian seasoning, salt, and pepper.
Cut a horizontal slit through the thickest portion of each chicken breast half to form a pocket. Stuff 1/4 cup mushroom mixture into each pocket. Close opening with a wooden pick.
Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add chicken; cook 2 minutes on each side or until browned. Stir in broth and tomatoes; bring to a boil. Cover, reduce heat, and simmer 12 minutes or until chicken is done. Remove chicken from pan. Cook sauce until reduced to 1 cup (about 5 minutes).