1. Heat oil in a large skillet over medium heat. Add onions, 1/4 teaspoon salt, and pepper to pan; cook 12 minutes, stirring frequently. Cover, reduce heat, and cook 8 minutes, stirring occasionally. Uncover and cook 5 minutes or until golden, stirring occasionally. Cool slightly. Combine onion mixture, 1/4 teaspoon salt, cheese, parsley, milk, and thyme in a small bowl, stirring with a fork.
2. Cut a horizontal slit through thickest portion of each chicken breast half to form a pocket; stuff 1 1/2 tablespoons cheese mixture into each pocket. Sprinkle chicken evenly with remaining 1/4 teaspoon salt.
3. Return pan to medium-high heat. Coat pan with cooking spray. Add chicken to pan; sauté 5 minutes; turn chicken over. Cover, reduce heat, and cook 10 minutes or until chicken is done.
Remove chicken from pan; let stand 10 minutes. Add wine to pan; bring to a boil, scraping pan to loosen browned bits. Cook until reduced by half (about 2 minutes). Add broth, and cook until reduced to 1/4 cup (about 9 minutes). Serve with chicken.