Prepare the breading: In a shallow bowl, whisk together the flour, grated Parmesan, garlic powder, 1/2 teaspoon salt, and 1/2 teaspoon pepper.
Prep the chicken: Cut each chicken breast into 3 pieces. Place each piece between sheets of plastic wrap and pound to about 1/4-inch thickness with a meat mallet or rolling pin.
Coat the chicken: Rub the chicken pieces with 1 tablespoon olive oil (divide between both sides). Dredge each piece in the flour-Parmesan mixture, shaking off excess.
Cook the first batch: Heat 2 tablespoons olive oil and 1 tablespoon butter in a large skillet over medium-high heat until shimmering. Add half the chicken and cook until golden, about 3–4 minutes on the first side, then 1–2 minutes on the second side. Transfer to a plate, season lightly with salt to taste, and cover with foil to keep warm.
Cook the second batch: Repeat with the remaining chicken, adding more oil or butter if needed, and transfer to the plate when done.
Make the sauce: Pour off excess fat from the skillet and wipe lightly with a paper towel. Add 1 teaspoon olive oil and the minced garlic and sauté about 20 seconds until fragrant. Add the chicken stock, lemon juice, lemon zest, and drained capers, and simmer 3–4 minutes until slightly reduced.
Finish the sauce: Reduce heat to low, add 2 tablespoons butter, and swirl until the sauce is glossy and slightly thickened. Stir in the chopped parsley, if using.
Serve: Arrange the cooked chicken on plates, spoon the lemon-caper sauce over the top, and garnish with extra parsley if desired. Serve immediately.