Heat oil in a large nonstick skillet over medium heat. Sprinkle fish with salt and pepper. Place fish in pan; cook 3 minutes on each side or until fish flakes easily when tested with a fork or until desired degree of doneness. Remove fish from pan.
Melt butter in pan. Add garlic; cook 1 minute or until lightly browned. Add broth, scraping pan to loosen browned bits. Bring to a boil; cook until reduced to 1/4 cup (about 6 minutes). Stir in minced capers and juice; cook 3 minutes or until sauce is slightly thickened. Remove from heat; stir in parsley. Serve immediately.
Lemon sole is an American variety of flatfish, which includes flounder. Most flatfish will work here.