Cut chicken breast in half lengthwise, season with salt and pepper, and dredge in flour.
In a large skillet over medium heat, melt 1 tablespoon of butter with 1 tablespoon of olive oil. Once sizzling, add two pieces of chicken and cook for 3-4 minutes without moving them for a golden crust.
Flip the chicken and cook for another 3-4 minutes until done. Transfer to a plate and repeat with remaining chicken, using more butter and oil as needed.
Drain excess oil from the skillet, add lemon juice, chicken broth, garlic, and capers. Bring to a boil, scraping up brown bits from the pan.
Adjust seasoning if necessary, then stir in remaining butter until emulsified.
Return chicken to the pan, coat it in the sauce, and warm for 1-2 minutes.
Garnish with fresh parsley and serve warm.
