How to Make It
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.