Melt butter in a 3-quart saucepan over medium-high. Whisk in flour and cook until thickened. Gradually whisk in whole milk. Bring to a boil, and cook, whisking constantly, until thickened, for 1 to 2 minutes. Whisk in grated Parmesan cheese, fresh lemon juice, salt, and black pepper. (If too thick, add more milk.) Serve immediately with chicken.