Season the chicken with 1 teaspoon salt and 1/2 teaspoon pepper. Heat 1 tablespoon of the oil in a 12-inch heavy-duty skillet over medium-high heat. When hot, add as many cutlets as will fit in a single layer and cook until lightly browned on both sides and just cooked through, 1 to 3 minutes per side. Shingle the chicken on a platter and tent with foil. Repeat with the remaining chicken, adding another tablespoon of oil between batches if the pan seems dry.

Reduce the heat to medium and add the capers, garlic, and the remaining 1 tablespoon oil to the pan. Cook, stirring, until the garlic softens and becomes fragrant, about 1 minute. Add the chicken broth and lemon juice, raise the heat to high, and cook, stirring, until the mixture reduces by about half, 3 to 4 minutes. Off the heat, swirl in half of the parsley and the butter and season with more lemon juice, salt, and pepper to taste. Drizzle the sauce over the chicken breasts, sprinkle with the remaining parsley, and serve immediately.