Cut 1 lemon in half lengthwise. Trim ends from 1 of the halves and slice crosswise ⅛ to ¼ thick; set aside. Squeeze juice from the other whole lemon and the remaining half lemon into small bowl. You should have about ¼ cup juice. Set aside.
Adjust your oven rack to middle position and heat to 200 degrees. Halve each chicken breast horizontally, then cover cutlets with plastic wrap and pound to an even ¼-inch thickness (or close) with meat pounder. Spread ½ cup flour in a shallow plate or dish.
Season cutlets with salt and pepper. Working with 1 cutlet at a time, dredge in flour, shaking off excess, and transfer to a large platter.
Heat 2 Tablespoons oil in a 12-inch skillet over medium-high heat until shimmering. Place 3-4 cutlets in the skillet, depending on how many fit without overcrowding, and cook until golden brown on first side, about 3 minutes. Using tongs, flip cutlets, reduce heat to medium, and cool until no longer pink and lightly browned on second side, about 2 minutes longer; transfer to large ovensafe platter or dish. Repeat with remaining cutlets, adding 2 Tablespoons more with each addition. Tent the chicken loosely with aluminum foil and transfer to oven to keep warm while making sauce.
Add 1 minced garlic to now-empty skillet and cook over medium heat about 30 seconds. Stir in 1 cup broth and lemon slices, scraping up any browned bits. Bring to simmer and cook until reduced to ⅓ cup, about 4 minutes.
Stir in 2 Tablespoons drained capers and lemon juice, bring to simmer, and cook until reduced to ⅓ cup again, about 1 minute. Turn off heat, whisk in 3 Tablespoons butter, 1 piece at a time. Stir in 2 Tablespoons parsley. Pour sauce over chicken and serve immediately.