1. Preheat oven to 350°.
2. Pulse bread in a food processor to form fine crumbs. Place in a dish.
3. Heat a large ovenproof skillet over medium-high heat. Coat pan with cooking spray. Add onions and mushrooms; sauté 10 minutes. Stir in thyme and garlic. Cool mixture 10 minutes; stir in cheese, 1/4 teaspoon salt, and 1/4 teaspoon pepper.
4. Cut a slit through thickest portion of each breast half to form a pocket. Stuff each with 1/4 cup mushroom mixture. Sprinkle chicken with remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Place flour in a shallow dish. Place eggs in a shallow dish. Dredge chicken in flour. Dip in eggs; dredge in breadcrumbs.
5. Heat oil in skillet over medium-high heat. Add chicken; cook 2 minutes on each side or until browned. Place pan in oven. Bake at 350° for 15 minutes or until a thermometer registers 165°.