If making the lemon butter sauce, prepare it first (see notes) and set aside.
In a shallow bowl, whisk together the flour, grated Parmesan, 1/2 teaspoon salt, 1/2 teaspoon pepper, paprika, and garlic powder.
Cut each chicken breast into three pieces to make six pieces total. Place a piece between sheets of plastic wrap and pound to about 1/4-inch thickness; repeat with remaining pieces.
Rub about 1 tablespoon olive oil evenly over the chicken pieces (divided among them), then dredge each piece in the flour-Parmesan mixture until fully coated.
Heat 2 tablespoons olive oil and 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add half the chicken pieces and cook 3–4 minutes on the first side until golden, then flip and cook 1–2 minutes more. Transfer to a plate and tent with foil.
Repeat with the remaining chicken, adding more oil if needed, and transfer cooked pieces to the plate; keep covered.
Thinly slice 1 lemon. Add the lemon slices to the hot skillet and cook 1 minute until softened, then flip and cook another 30 seconds. Remove and set aside.
Add the remaining 1 tablespoon butter to the skillet and increase heat to high. Add the green beans with a pinch of salt and some black pepper. Toss to coat, let cook undisturbed for 3 minutes to char, then stir occasionally 1–3 minutes more until crisp-tender.
Remove the pan from the heat. Nestle the cooked chicken and lemon slices on top of the green beans, drizzle with fresh lemon juice from the remaining lemon, and sprinkle additional Parmesan if desired. Serve with the optional lemon butter sauce.
