Pat chicken breasts dry with paper towels. Use a sharp knife to cut each breast in half horizontally to create two thinner pieces.
Place each piece of chicken between two sheets of plastic wrap. Use a meat mallet or rolling pin to gently pound the chicken to an even thickness of about ¼ inch.
On a wide, flat plate, mix together flour, kosher salt, and black pepper. Dredge each chicken piece in the flour mixture, coating all sides.
In the same skillet over medium heat, heat 1 tablespoon olive oil and 1 tablespoon unsalted butter until sizzling.
Working in batches, add four chicken pieces to the skillet. Cook for about 6 minutes per side, or until golden brown and the internal temperature reaches 165°F.
Transfer cooked chicken to a plate and loosely cover with foil to keep warm.
Add the remaining olive oil and butter to the skillet and cook the remaining chicken pieces the same way. Transfer to the same plate.