Butterfly the chicken breasts: slice horizontally through the middle to create 4 thin cutlets. Pound to an even ½ inch thickness.
Season chicken with salt and pepper. Dredge in flour, shaking off excess.
In a large skillet over medium high heat, melt 2 tbsp butter with 3 tbsp olive oil. Add 2 chicken cutlets and cook for 3 mins per side until golden. Remove to a plate. Repeat with remaining 2 tbsp butter, 2 tbsp oil, and last 2 cutlets.
Remove skillet from heat. Add lemon juice, chicken stock, and capers. Return to heat, bring to a boil, scraping up brown bits. Boil for 1 minute.
Reduce heat to low. Return all chicken to the skillet and simmer for 2-3 minutes. Remove chicken to a platter.
Turn off heat. Add remaining 2 tbsp butter to the sauce and whisk until melted and smooth.
Pour sauce over chicken, garnish with parsley, and serve.