Broil the bell peppers until blackened and blistered. Let cool, then peel, seed and slice into fat strips.

In a small skillet, melt the butter over medium heat. Add the onion and garlic and cook until softened, about 5 minutes. Pour in the vermouth (or white wine) and stir up any browned bits. Let cool.

Place the ground meat in a bowl; season with salt and pepper. Mix in the onion-garlic mixture, the sweet paprika, marjoram, chili powder and allspice. Form 4 large patties, making them thinner at the centers and thicker at the edges for even cooking.

In a large skillet, heat the olive oil, 1 turn of the pan, over medium-high heat. Add the patties and cook, turning once, for 8 minutes for medium. Add the cheese in the last minute of cooking, tenting loosely with foil to melt.

In a small bowl, make a sauce by stirring together the sour cream, ketchup, pepperoncini brine, hot sauce, salt and pepper.

On the roll bottoms, layer the patties and roasted peppers. Slather the sauce on the roll tops and set in place.