In a large cast-iron skillet over medium heat, melt 1 tablespoon butter. Add sliced onions; cook, stirring occasionally, for about 25 minutes, until wilted and caramelized (turn heat down if they start to burn). Add vinegar and stir for a few more minutes. Remove onions from skillet and set aside.
While onions are finishing up, work a teaspoon of the Worcestershire sauce into the ground beef and form 2 thin patties to fit the size and shape of the bread slices. Cook patties in the now-empty skillet over medium-high, 3 minutes per side or until cooked through. Set patties aside with the onions.
Butter the outsides of the bread and place butter-side down in skillet. Split cheese across the top of the bread slices, then top with caramelized onions and patties. Cover skillet with lid until bread is toasted and cheese is melted. Serve immediately.
Per serving: 715 calories, 47 grams total fat (23 grams saturated fat), 38 grams total carbohydrates, 144 milligrams cholesterol, 932 milligrams sodium, 5 grams fiber, 33 grams protein.
