Cut each chicken breast in half crosswise, then cut the 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. Then, place cutlets on a rimmed sheet tray and season both sides with Kosher salt and black pepper. Set aside for 15 minutes.

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.

Prepare dredging station with flour in a shallow dish. Place a wire rack set in a sheet tray. Dredge cutlets in flour, shaking gently to remove excess. Place on wire rack. Repeat with all cutlets.

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 each side. About 2 to 3 minutes each side. Return cutlets to wire rack. Repeat with 2 tablespoons oil and remaining 6 cutlets.

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 a simmer, scraping up any browned bits.

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. Turn off heat, whisk in butter. Stir in capers and parsley. Season with salt and pepper to taste. Spoon sauce over chicken and serve.