Serve with buttered pasta, white rice, potatoes, or crusty bread and a simple steamed vegetable.
1. Cut each chicken breast in half crosswise, then cut thick half in half again horizontally, creating 3 cutlets of similar thickness. Place cutlets between sheets of plastic wrap and gently pound to even ½-inch thickness. Place cutlets in bowl and toss with 1 teaspoon salt and ¼ teaspoon pepper. Set aside for 15 minutes.
2. Halve lemon lengthwise. Trim ends from 1 half, halve lengthwise again, then slice crosswise ¼ inch thick; set aside. Juice remaining half and set aside 1½ tablespoons juice.
3. Spread flour in shallow dish. Working with 1 cutlet at a time, dredge cutlets in flour, shaking gently to remove excess. Place on wire rack set in rimmed baking sheet. Heat 2 tablespoons oil in 12-inch skillet over medium-high heat until smoking. Place cutlets in skillet, reduce heat to medium, and cook until golden brown on 1 side, 2 to 3 minutes. Flip and cook until golden brown on second side, 2 to 3 minutes. Return cutlets to wire rack.
4. Add remaining 1 teaspoon oil and shallot to skillet and cook until softened, 30 seconds. Add garlic and cook until fragrant, 30 seconds. Add broth, reserved lemon juice, and reserved lemon slices and bring to simmer, scraping up any browned bits.
5. Add cutlets to sauce and simmer for 4 minutes, flipping halfway through simmering. Transfer cutlets to platter. Sauce should be thickened to consistency of heavy cream; if not, simmer 1 minute longer. Off heat, whisk in butter. Stir in capers and parsley. Season with salt and pepper to taste. Spoon sauce over chicken and serve.