Place the chicken cutlets between two pieces of plastic wrap. Using a meat mallet or a heavy rolling pin, gently pound the chicken to an even ¼-inch thickness. Season both sides generously with salt and pepper. Place the flour in a shallow dish and lightly dredge each cutlet, shaking off any excess.

In a large skillet over medium-high heat, add one tablespoon of olive oil and one tablespoon of butter. Once the butter is melted and foaming, carefully place two chicken cutlets in the pan. Do not overcrowd. Cook for about 2-3 minutes per side, until golden brown and cooked through. Transfer the cooked chicken to a plate and repeat with the remaining cutlets.

Reduce the heat to medium. Add the minced garlic to the skillet and cook for just 30 seconds until fragrant. Pour in the white wine to deglaze, scraping up all the browned bits from the bottom of the pan with a wooden spoon. Let the wine reduce by about half.

Pour in the chicken broth and fresh lemon juice, then add the drained capers. Bring the sauce to a simmer and let it cook for about 3-4 minutes, allowing it to reduce slightly. Taste and adjust seasoning with salt and pepper if needed.

Remove the skillet from the heat. Add the remaining 3 tablespoons of cold butter and the fresh parsley. Swirl the skillet continuously until the butter has melted completely, creating a smooth, glossy sauce. Return the chicken cutlets to the pan, spooning the sauce over the top to rewarm them.