Place each chicken breast half between 2 sheets of heavy-duty plastic wrap; pound to 1/2-inch thickness using a meat mallet or rolling pin. Sprinkle chicken evenly with salt and pepper.
Heat oil and butter in a large nonstick skillet over medium heat. Add chicken; cook 6 minutes on each side or until done. Remove chicken from pan; keep warm.
Add shallots to pan; cook 1 minute, stirring constantly. Add bourbon to pan; bring to a boil, scraping pan to loosen browned pits. Cook until bourbon mixture is reduced to 2 tablespoons (about 5 minutes). Add broth and barbecue sauce; cook until broth mixture is reduced to 1/2 cup (about 3 minutes). Serve sauce over chicken.