Pat the chicken breasts dry and slice each breast in half lengthwise to make 4 thin cutlets. Season both sides with the kosher salt and ground black pepper.
Place the all-purpose flour in a wide bowl. Dredge each cutlet in the flour, shaking off any excess.
Heat the olive oil and 2 tablespoons of the butter in a large nonstick skillet over medium heat until the butter melts and the pan is hot.
Add the floured chicken cutlets and cook 3–4 minutes per side, until golden brown and cooked through. Remove the chicken from the skillet and set aside.
Reduce heat to low and add the remaining 4 tablespoons butter to the skillet. When melted, add the chopped garlic and cook 30–60 seconds until fragrant, stirring so it doesn't burn.
Stir in the minced parsley, return the chicken to the skillet, and spoon the garlic-butter sauce over the cutlets to coat and heat through.
Remove from heat, sprinkle the grated Parmesan over the chicken, and serve with lemon wedges if desired.
