Step 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 2 teaspoons salt and ½ teaspoon pepper. Set aside for 15 minutes.

Step 2

Halve 1 lemon lengthwise. Trim ends from 1 half, halve lengthwise again, then cut crosswise ¼-inch-thick slices; set aside. Juice remaining half and whole lemon and set aside 3 tablespoons juice.

Step 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 6 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. Repeat with 2 tablespoons oil and remaining 6 cutlets.

Step 4

Add remaining 1 teaspoon oil and shallot to skillet and cook until softened, 1 minute. 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.

Step 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.