Beat together egg and salt and pepper to taste in a bowl until well blended. Add ground chicken and mix until just combined. Form into 4 patties about 3/4 inch thick. Combine 1/4 cup flour and salt and pepper to taste. Dredge patties in seasoned flour. Melt butter in oil in a heavy nonstick skillet over medium high heat. Cook burgers about 5 minutes per side, turning once, until golden brown and chicken is just cooked through. Drain burgers on paper towels and keep warm. Sprinkle 2 Tbs. Flour into drippings in skillet over medium heat. Cook 2-3 minutes, whisking until lightly browned. Whisk in milk and hot pepper sauce. Increase heat to medium high and stir constantly 2-3 minutes, or until thickened. Season with salt and pepper to taste and serve over chicken burgers.
