1 tablespoon unsalted butter, plus more for brushing
2 medium shallots, thinly sliced
1 cup dry red wine
1 tablespoon honey
1 teaspoon red wine vinegar
Pinch of herbes de Provence
Pinch of crushed red pepper
14 ounces ground chuck
14 ounces ground short ribs (ask your butcher to do this)
Vegetable oil, for brushing
4 brioche burger buns, split
8 slices of deli-quality American cheese (about 4 ounces)
In a medium saucepan, melt the 1 tablespoon of butter. Add the shallots and cook over moderate heat, stirring occasionally, until lightly browned and tender, 5 minutes. Add the wine, honey, vinegar, herbes de Provence and crushed red pepper and cook over moderately low heat, stirring occasionally, until a thick jam forms, about 15 minutes. Season with salt.
Meanwhile, lightly mix the ground meats. Divide into 4 mounds and pack loosely into 5-inch patties. Season generously with salt and make a slight depression in the center of each burger to help the patties retain their shape while they cook.
Light a grill or preheat a grill pan; brush with oil. Brush the cut sides of the buns with butter and grill until toasted. Grill the burgers over high heat until browned and crusty, about 4 minutes. Flip the burgers and cook for 2 minutes longer. Top with the cheese, cover the grill and let the cheese melt, about 1 minute. Transfer the burgers to the buns and top with the shallot jam. Close the burgers and serve.
The shallot jam can be refrigerated for up to 5 days. Bring to room temperature before serving.