Cut chicken breast in half lengthwise (butterflied). Season with salt and pepper. Dredge each piece in flour and shake off excess.
In a large skillet over medium heat, melt 1 tablespoons of butter with 1 tablespoons olive oil. When butter and oil start to sizzle, add 2 pieces of chicken and cook for 3-4 minutes, without moving the chicken. When chicken is browned, flip and cook other side for 3 -4 minutes or until cooked through.
Remove chicken and transfer to plate. Melt 1 more tablespoons butter and add another 1 tablespoons olive oil. When butter and oil start to sizzle, add the other 2 pieces of chicken and repeat. When cooked though, transfer chicken to the plate and remove pan from heat.
Drain used oil from pan. Add the lemon juice, chicken broth, garlic, and capers to the pan. Return pan to heat and bring to a boil, scraping up brown bits from the pan. Check for seasoning and adjust as needed. Add remaining 2 tablespoons butter to sauce and stir until emulsified. Add chicken back to pan, garnish with parsley and serve.