Start by placing the chicken breasts between two sheets of plastic wrap. Use a meat mallet or rolling pin to gently pound the chicken to even thickness, about ½ inch thick. Season both sides with kosher salt and black pepper.
In a shallow bowl, add the all-purpose flour. Lightly dredge each chicken breast in the flour, shaking off any excess.
In a large skillet over medium heat, add the olive oil and 2 tablespoons of unsalted butter. Once the butter has melted and is sizzling, carefully place the dredged chicken breasts in the skillet. Cook for about 5-7 minutes on each side, until golden brown and cooked through. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining 4 tablespoons of unsalted butter. Once melted, add the finely chopped garlic and sauté for about 1 minute until fragrant.
Stir in the minced parsley and grated Parmesan cheese, allowing the cheese to melt into the sauce. Return the chicken to the skillet, spooning the garlic butter sauce over each piece. Cook for an additional minute to warm the chicken through. Serve with lemon wedges for an extra burst of flavor.