Cut each chicken breast in half lengthwise to make 4 thin cutlets; season both sides with kosher salt and freshly ground black pepper.
Put the flour in a shallow dish and dredge each chicken piece, shaking off excess flour.
Heat a large skillet over medium heat and melt 1 tablespoon butter with 1 tablespoon olive oil until sizzling.
Add two chicken cutlets to the skillet and cook undisturbed for 3–4 minutes until browned; flip and cook another 3–4 minutes until cooked through. Transfer to a plate.
Wipe out excess oil if needed, then melt 1 tablespoon butter with 1 tablespoon olive oil in the skillet. Cook the remaining two cutlets the same way, 3–4 minutes per side, then transfer to the plate.
Pour off any excess fat from the pan, then add the lemon juice, chicken broth, minced garlic, and drained capers. Return the pan to medium-high heat and bring to a boil, scraping up browned bits from the bottom.
Adjust seasoning if needed, then stir in the remaining 2 tablespoons butter until the sauce is emulsified and slightly thickened.
Return the chicken to the pan to warm in the sauce for 1 minute, spoon sauce over the cutlets, garnish with chopped parsley, and serve.