Season the chicken with salt and pepper on both sides. Dredge the cutlets in flour and shake off the excess. Place the dredged cutlets on a parchment paper-lined baking sheet.

Heat a large nonstick pan to a touch less than medium heat. Add 1 tablespoon of olive oil and butter. Sear the chicken cutlets until cooked through (about 3-4 minutes per side). Set aside on a platter lightly tented with foil. Do not crowd the pan, work in batches, adding more olive oil and butter as needed.

Using the same pan, turn the heat to low. Add garlic and a touch more olive oil. Sauté until golden (about 1-2 minutes), then add the capers and cook for another minute.

Roll the remaining butter cubes in the leftover flour and shake off the excess. Add one cube at a time to the pan. Stir the sauce to emulsify with a whisk or wooden spoon. Taste and adjust salt, pepper, and lemon juice levels as necessary.

Add the chicken and half of the parsley to the sauce. Gently coat all the pieces and cook for 2-3 minutes to warm through. Turn off heat and add the remaining parsley. Serve with the remaining lemon wedges.