1. Preheat broiler to high.
2. Combine first 4 ingredients in a small bowl. Cut a horizontal slit through thickest portion of each breast half to form a pocket. Stuff 2 tablespoons cheese mixture into each pocket; close opening with a wooden pick. Sprinkle chicken with 1/4 teaspoon salt.
3. Heat a large nonstick skillet over medium-high heat; add 1 tablespoon olive oil to pan. Add chicken to pan, and cook for 2 minutes on 1 side or until browned. Arrange chicken on a baking sheet (browned side up); broil for 8 minutes or until done.
4. Heat skillet over medium-high heat. Add remaining 1 teaspoon oil, red pepper, and spinach; cook for 2 minutes or until spinach wilts. Stir in the remaining 1/4 teaspoon salt.