Adapted from Susan Herrmann Loomis, as featured in French Grill: 125 Refined & Rustic Recipes.
Earthy and rich upgrade to the basic cheeseburger.
Using your hands, gently mix beef with salt, pepper, and allspice. Form meat into 4 patties, slightly concave in the center and slightly wider than bun (they will shrink up). Chill patties.
Cut Camembert into 1⁄4-in. slices, then quarter slices crosswise.
Meanwhile, heat a grill to high (500° to 550°). (** see Recipe Notes) Set a medium cast-iron skillet on grill and add 1 tablespoon butter and 1/2 tablespoon oil; when butter melts, add the sliced onions. Stir, season lightly with salt, cover, and cook until onions soften and turn golden, about minutes, stirring every couple of minutes. Scoop onions into a bowl.
Add another 1 tablespoon butter and 1/2 tablespoon oil to skillet, then add the sliced apples. Cook, covered, until golden, about 3 minutes. Stir and cook until tender throughout, about 8 minutes more. Remove pan from heat, season lightly with salt, and set aside.
Reduce heat to medium-high (400° to 450°). Grill the burgers until browned, about 4 minutes. Turn and cook 2 minutes more, then top each with 2 or 3 slices cheese, cover grill, and let melt 1-2 minutes. Transfer to a baking sheet.
Toast the buns, cut side down, about 2 minutes. Transfer to another baking sheet.
Assemble burgers: Slather bottom sides of buns with mayonnaise if you like. Divide apples among bottom halves of buns. Top with burgers, cheese side up, then the onions. Sprinkle with parsley if you like, then add bun tops. Enjoy immediately.
** Alternatively, the onion and apple slices can be sautéed on the stovetop in a medium-size saucepan.