Prepare the Chicken Cutlets: Place a chicken breast inside a large zip-top bag or between two pieces of plastic wrap. Using a meat mallet or rolling pin, pound the chicken to an even ¼-inch thickness. Repeat with the remaining chicken, then slice each pounded breast in half to create four cutlets. Season both sides generously with salt and pepper.

Dredge and Pan-Sear the Chicken: Spread the all-purpose flour on a shallow plate. Lightly press each seasoned chicken cutlet into the flour, coating both sides, then gently shake off any excess. In a large skillet over medium-high heat, melt 2 tablespoons of butter with 1 tablespoon of olive oil. Once foaming, carefully place two chicken cutlets in the pan, avoiding overcrowding. Cook for 2-3 minutes per side, until golden brown and cooked through. Transfer to a clean plate and repeat with the remaining cutlets.

Build the Pan Sauce: Lower the heat to medium. Add the minced garlic to the skillet and cook for about 30 seconds until fragrant. Pour in the white wine to deglaze, scraping up any browned bits from the bottom of the pan with a wooden spoon. Let the wine simmer and reduce by about half.

Simmer and Finish: Pour in the chicken broth and fresh lemon juice, bringing the mixture to a simmer. Add the drained capers and let the sauce cook for another 2-3 minutes to allow the flavors to meld. Taste and season with salt and pepper if needed.

Bring It All Together: Turn the heat to low. Return the seared chicken cutlets and any accumulated juices to the skillet. Add the remaining 2-3 tablespoons of cold butter. Gently swirl the skillet until the butter has melted and emulsified, creating a glossy sauce. Do not let the sauce boil after adding the butter, or it may separate. Garnish with fresh parsley and serve immediately.