Preheat oven to 200°F. Sprinkle chicken with salt and pepper and dredge in flour. Warm 1 Tbsp. butter and 2 Tbsp. oil in a large skillet over medium-high heat until butter foams. Add 2 chicken breast halves; cook until browned, 3 minutes. Flip chicken; cook until firm and browned on other side, 3 minutes longer. Transfer to a platter and loosely cover with foil.
Add 1 Tbsp. butter and 1 Tbsp. oil to skillet and repeat with remaining chicken. Add to platter, cover and place in oven.
Add lemon juice and wine to skillet and bring to a boil, scraping up browned bits from bottom of pan. Boil, stirring occasionally, until thickened, 5 minutes. Remove from heat; stir in capers and remaining 1 Tbsp. butter. Cook, stirring, until butter melts. Season with salt and pepper. Pour sauce over chicken and serve.