Place each chicken breast between 2 sheets of heavy-duty plastic wrap. Pound breasts to 1/2-inch thickness using a meat mallet or small heavy skillet.

Spread flour into a wide, shallow dish. Press chicken into flour to coat completely.

Melt butter with olive oil in a large skillet over medium-high heat; cook chicken in hot butter mixture until no longer pink in the center and the juices run clear, 3 to 5 minutes per side. Remove chicken from skillet to a serving platter and cover with a lid to keep warm.

Pour white wine, lemon juice, capers, and garlic into the pan. Bring the liquid to a boil while scraping the browned bits of food from the bottom of the pan with a wooden spoon; cook at a boil until slightly thickened, about 2 minutes. Season the mixture with salt and pepper.

Drizzle sauce over and around the chicken; garnish with parsley.