In a skillet over medium heat; warm oil and butter.
Add onions, salt and pepper.
Cook until onions are light golden in color and caramelized, 20-25 minutes.
In a small mixing bowl; soak bread in milk. Reserve.
In a mixing bowl; combine beef and egg.
Stir in bread mixture, Worcestershire sauce, catsup, mayonnaise, oyster sauce, onion, salt and pepper.
Form into 8 patties. In a large skillet over high heat; hot oil.
Brown patties until cooked through, about 15 minutes, turning only when hamburger is cooked 3/4 of the way.
Covering helps patties cook through and hold form.
Transfer to a plate; keep warm. Into skillet of onions; sprinkle flour over onions, stir to blend into onions.
Remove from heat; seat aside.
Into hamburger skillet; pour wine into; scraping any browned bits form bottom of pan.
Transfer onions into skillet.
Stir in broth until thickened.
Return hamburger patties to skillet. Heat through.