Preheat oven to 200°F. Sprinkle chicken with salt and pepper; dredge in flour.
Warm 1 Tbsp. butter and 1 Tbsp. vegetable oil in a large skillet over medium-high heat until butter foams. Add half of cutlets and cook until browned, about 4 minutes. Turn and cook until firm and browned on both sides, about 3 minutes longer. Transfer to a plate and loosely cover with foil to keep warm.
Add another 1 Tbsp. each butter and oil to skillet and repeat with remaining cutlets. Add to platter and keep warm in oven.
Add Worcestershire sauce, vinegar and chicken broth to empty skillet and bring to a boil, scraping up browned bits from bottom of pan with a wooden spoon. Boil, stirring occasionally, until thickened, about 5 minutes. Remove from heat and stir in remaining 1 Tbsp. butter until it is melted. Season sauce with salt and pepper. Pour sauce over chicken and garnish with chopped parsley. Serve immediately.