1. Place each chicken breast half between 2 sheets of heavy-duty plastic wrap; pound to 1/2-inch thickness using a meat mallet or small heavy skillet. Place 1 teaspoon flour in a small bowl, and place remaining flour in a shallow dish. Sprinkle both sides of chicken evenly with salt and pepper. Dredge chicken in flour in shallow dish; shake off excess.
2. Melt 1 tablespoon butter in a large skillet over medium-high heat. Add 1 tablespoon oil to pan; swirl to coat. Add chicken to pan; sauté 4 minutes on each side or until done. Remove chicken from pan; keep warm.
3. Heat remaining 1 tablespoon oil in pan; swirl to coat. Add shallots to pan; sauté 3 minutes, stirring frequently. Add garlic; sauté 1 minute, stirring constantly. Add wine; bring to a boil, scraping pan to loosen browned bits. Cook until liquid almost evaporates, stirring occasionally. Add 1/4 cup broth to reserved 1 teaspoon flour; stir until smooth. Add remaining 1/2 cup broth to pan; bring to a boil. Cook until reduced by half (about 5 minutes). Stir in flour mixture; cook 1 minute or until slightly thick, stirring frequently. Remove from heat; stir in remaining 1 1/2 tablespoons butter, juice, and capers. Place 1 chicken breast half on each of 4 plates; top each serving with about 2 tablespoons sauce. Sprinkle each serving with about 2 teaspoons parsley.