Heat oven to 375°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.
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.
Let chicken rest 10 minutes. Peel breast meat away from bone and slice.