Butterfly each chicken breast by cutting in half lengthwise; season both sides with kosher salt and freshly ground black pepper, then dredge in all-purpose flour and shake off excess.
Heat 1 tablespoon butter and 1 tablespoon olive oil in a large skillet over medium heat until sizzling.
Add two pieces of floured chicken to the skillet and cook without moving for 3–4 minutes, until the underside is golden brown; flip and cook the other side 3–4 minutes more, until cooked through.
Transfer cooked chicken to a plate and repeat with 1 tablespoon butter and 1 tablespoon olive oil, cooking the remaining chicken pieces the same way; transfer to the plate when done.
Pour off any excess oil from the skillet, then add the lemon juice, chicken broth, minced garlic, and drained capers; return the pan to medium-high heat and bring the sauce to a boil, scraping up any browned bits from the bottom.
Taste the sauce and adjust seasoning if needed, then stir in the remaining 2 tablespoons butter until the sauce is emulsified and glossy.
Return the chicken to the skillet to warm in the sauce for a minute, spoon sauce over the pieces, garnish with chopped parsley, and serve.