Pat chicken pieces dry with paper towels. In a shallow bowl, combine flour, salt, and pepper. Toss chicken in flour mixture until lightly coated, shaking off excess.

Heat olive oil and 2 tablespoons butter in a large skillet over medium-high heat. Add chicken in a single layer and cook undisturbed for 4 minutes until golden brown. Flip and cook 3 minutes more until cooked through. Transfer to a plate.

Reduce heat to medium. Add remaining 2 tablespoons butter to the skillet. Once melted, add garlic and cook 30 seconds until fragrant but not browned. Pour in wine and broth, scraping up any browned bits. Simmer 3-4 minutes until slightly reduced. Stir in lemon zest, lemon juice, and red pepper flakes if using.

While sauce simmers, bring a large pot of salted water to a boil. Cook linguine 1 minute less than package directions for al dente. Reserve 1 cup pasta water, then drain.

Add pasta to the skillet with sauce, tossing to coat. Add chicken and any accumulated juices. Cook 1-2 minutes, adding reserved pasta water a splash at a time until sauce clings to noodles. Remove from heat and stir in parsley and Parmesan. Serve immediately with extra cheese on the side.