Place both meats in bowl of mixer; blend together. Add pepper, cayenne and Worcestershire sauce. Season with salt and blend again. Set aside.
Heat half of oil in saucepan over medium heat. Add both onions, seasonings and garlic. Stir and cook 3 minutes over medium heat.
Add cooked onions to meat mixture. Blend to incorporate. Add egg and blend again.
Shape mixture into hamburgers and brush with oil.
Preheat barbecue at medium.
Oil grill and add hamburgers. Cover and cook 5 minutes on each side. Season to taste. Serve with condiments.