Preheat your oven to 375 degrees F. Season the chicken on both sides with salt and pepper.
Heat a large saute pan over medium-high heat and add a drizzle of olive oil and 1 tablespoon of the butter. Add the chicken pieces, skin-side down, and cook until golden brown, 3 to 5 minutes. Flip and cook on the other side until golden brown, 2 to 3 minutes, then remove to a sheet tray. Place the chicken in the oven and cook until the internal temperature reaches 165 degrees F on an instant-read thermometer, 15 to 20 minutes.
Place the saute pan back on the heat over medium heat and add 2 tablespoons of the butter along with the garlic and a pinch of salt. Cook until the garlic softens and becomes aromatic, 1 to 2 minutes. Next, add the white wine and reduce slightly, 2 to 3 minutes, before adding the chicken stock.
Bring the liquid to a strong simmer, 2 to 3 minutes, then add the capers and lemon juice. Remove from the heat and swirl in the remaining 1 tablespoon of butter and the parsley. Place the chicken pieces back in the pan or plate the chicken and spoon the sauce over the top.