Adjust oven rack to middle position and heat oven to 200℉. Halve 1 lemon, trim ends, and slice lemon into thin half-moons. With the remaining lemon, squeeze ¼ cup juice.
Cut chicken horizontally into 2 thin cutlets, then cover with plastic wrap and pound to ½-inch thickness.
Pat cutlets dry with paper towels. Season with salt and pepper.
Dredge chicken in flour, shaking off any excess.
Heat 2 tablespoons oil in large skillet over medium-high heat. Heat oil until just smoking.
Brown half of the chicken cutlets until lightly golden on both sides (about 4 minutes).
Transfer to plate and keep warm in the oven. Repeat with remaining oil and cutlets.
Add shallot and garlic to the oil left in the skillet and cook over medium heat until shallot is softened (about 2 minutes).
Stir in broth and lemon slices. Scrape up any browned bits and simmer until reduced and slightly syrupy (about 8 minutes).
Stir in lemon juice and capers. Turn the heat to low and whisk in putter, 1 piece at a time.
Turn off heat, stir in parsley, and season with salt and pepper.
Spoon sauce over chicken and serve.
Notes
For even cooking, ensure the chicken breasts are of uniform thickness. If needed, place them between plastic wrap and gently pound with a meat mallet or the bottom of a pan.
Add extra flavor to the chicken by mixing the flour with a pinch of salt, pepper, and any other desired herbs or spices before dredging.
When adding broth to deglaze the skillet, scrape the bottom of the pan well to incorporate all the flavorful browned bits into the sauce.
To achieve a silky, emulsified sauce, remove the skillet from the heat before whisking in the butter, and add it gradually, whisking continuously.