If making the optional lemon butter sauce, prepare it first following the notes below.
In a shallow bowl, whisk together the flour, grated Parmesan, salt, pepper, paprika, and garlic powder.
Cut each chicken breast into 3 pieces. Place each piece between two sheets of plastic wrap and pound to about 1/4-inch thickness.
Rub a total of 1 tablespoon olive oil over the chicken pieces (divide between both sides), then dredge each piece in the flour-Parmesan mixture until evenly coated.
Heat 2 tablespoons olive oil and 1 tablespoon butter in a 12-inch 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 cover with foil. Repeat with remaining chicken, adding oil if needed.
Thinly slice 1 lemon. Add the lemon slices to the skillet and cook 1 minute until softened, flip and cook 30 seconds more, then remove and set aside.
Add the remaining 1 tablespoon butter to the skillet and increase heat to high. Add the green beans, a pinch of salt and some pepper, toss to coat, then let cook undisturbed 3 minutes to char. Stir occasionally and cook another 1–3 minutes until crisp-tender and slightly charred.
Remove the pan from heat, return the cooked chicken and lemon slices to the skillet on top of the green beans. Drizzle with fresh lemon juice from the second lemon and sprinkle additional Parmesan if desired, then serve with the lemon butter sauce if using.