1. Heat oven to 375 degrees F. In saucepan, melt 2 teaspoons butter over high heat. Stir in onion and cook 3 minutes. Stir in apple, apricot preserves, salt, and cinnamon. Cook 2 minutes, stirring frequently. Cool for 5 minutes.
2. Peel skin off chicken breast, leaving it attached at wing joint. Place half of apple mixture on each breast; fold skin back over filling and secure with toothpick. Melt remaining 1 teaspoon butter and brush over tops of chicken breasts. Place in baking pan, and bake 45 minutes.
3. Let chicken rest 10 minutes. Peel breast meat away from bone and slice.