Juice the lemon until you have 1/4 cup. Chop the parsley until you have 2 tablespoons.
Butterfly the chicken breasts and pound them to an even 1/4-inch thickness in a zip-top bag.
Season the chicken with 1 teaspoon salt and 1/2 teaspoon black pepper.
Dredge each piece in flour, shaking off the excess.
Heat 2 tablespoons olive oil in a large skillet over medium-high heat. Cook the chicken in a single layer for 2-3 minutes per side until golden brown. Repeat with the remaining chicken if needed.
Add white wine to the pan and cook until reduced by half, about 1 minute.
Stir in lemon juice, chicken broth, and remaining salt. Cook until slightly thickened, about 3 minutes.
Add butter and swirl until melted. Pour sauce over chicken and sprinkle with parsley.