Pour flour into a shallow dish. Dredge chicken through the flour, shaking off excess flour; season with salt and pepper.

Heat olive oil in a skillet over medium-high heat for 1 minute. Place a small piece of chicken into the hot oil. If the oil sizzles, it is ready. Cook chicken in the hot oil until golden brown on the outside and no longer pink in the center, about 5 minutes per side. Transfer chicken to a serving dish, retaining the oil and browned bits of food in the pan.

Lower heat to medium and pour apple juice and red wine into the pan; bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Let mixture sit over medium heat until sauce is a syrup-like consistency, 2 to 3 minutes. Stir butter until sauce until melted; season with salt and pepper. Spoon sauce over chicken.