Slice the 2 chicken breasts in half horizontally to create 4 thinner cutlets. Place each cutlet between two sheets of plastic wrap and use a meat mallet or rolling pin to pound them to an even 1/4-inch thickness. This ensures they cook quickly and evenly.

In a shallow dish, whisk together the all-purpose flour, grated Parmesan cheese, kosher salt, and black pepper.

Dredge each chicken cutlet in the flour mixture, shaking off any excess. Set the coated chicken on a clean plate.

In a large 12-inch skillet, heat 2 tablespoons of olive oil and 2 tablespoons of butter over medium-high heat. Once the butter is melted and foamy, carefully place two chicken cutlets in the pan. Be sure not to overcrowd the pan.

Sauté for 3-4 minutes per side, until golden brown and cooked through. Transfer the cooked chicken to a plate and keep warm. Repeat with the remaining two cutlets.

Reduce the heat to medium. Add the minced garlic to the same skillet and cook for about 30 seconds until fragrant, stirring constantly.

Pour in the white wine to deglaze the pan, scraping up any browned bits from the bottom with a wooden spoon. Let the wine simmer and reduce by about half, which should take 1-2 minutes.

Stir in the chicken broth, fresh lemon juice, and capers. Bring the sauce to a simmer and let it cook for 3-4 minutes to allow the flavors to meld and the sauce to reduce slightly.

Remove the skillet from the heat. Whisk in the remaining 4 tablespoons of cold, cubed butter, one piece at a time, until the sauce is creamy and emulsified. Stir in the chopped fresh parsley.

Return the chicken cutlets to the skillet, turning them to coat in the sauce. Serve immediately, spooning extra sauce and capers over the chicken.