In a large skillet, heat oil over medium heat. Add onion and peppers; cook and stir 5 minutes. Stir in beer. Bring to a boil. Reduce heat; simmer, uncovered, 15-20 minutes or until vegetables are tender and liquid is almost evaporated. Remove and keep warm.
In a small bowl, combine beef and bratwurst, mixing lightly but thoroughly. Shape into four 3/4-in.-thick patties.
In a shallow bowl, mix egg and milk. Place bread crumbs in a separate shallow bowl. Dip patties in egg mixture, then roll in crumb mixture to coat.
In the same skillet over medium heat, cook burgers 3-4 minutes on each side or until a thermometer reads 160° for medium doneness and juices run clear, topping with cheese during the last 1-2 minutes of cooking. Serve burgers on buns with mustard and onion mixture.