To make chicken cutlets, butterfly the chicken breasts by cutting them horizontally in half all the way through. (You will end up with 6 pieces total.) If they are still thick, use the flat side of a meat tenderizer to pound them to ¼-inch (6.4 mm) thickness. Pat the chicken dry with paper towels.

Season both sides of the chicken cutlets with salt and pepper. Place the grated parmesan into a shallow bowl. Dredge the chicken on both sides in the parmesan.

In a 12-inch skillet (I used this cast iron braiser because it heats well and can fit a lot of chicken cutlets at once, but a regular non-stick skillet works too), heat a tablespoon of olive oil over medium-high heat, until shimmering. Working in batches, add the chicken cutlets in a single layer. Saute for about 3 minutes per side, until browned. Transfer to a plate and cover with foil to keep warm. Repeat with all the chicken cutlets, adding the remaining tablespoon of olive oil halfway through.

Reduce heat to medium. Add a tablespoon of butter to the pan. Add minced garlic and saute for about a minute, until fragrant.

Add the chicken broth and lemon juice. Scrape the bottom of the pan with a wooden spoon or spatula to release any browned bits (this is called deglazing). Add the capers. Increase heat to bring to a boil, then simmer for 7-10 minutes, until volume is reduced by half and slightly thickened.

Stir in remaining butter, until melted. Simmer for another 3-5 minutes, until slightly reduced to your liking. Taste and adjust salt and pepper to taste, if desired.

Return the chicken to the pan and coat in sauce. Serve with fresh parsley, if desired.