Make the slaw: Combine the cabbage, apple, carrot and scallions in a bowl. Add 2 tablespoons dijonnaise and the vinegar; season with salt and pepper and toss. Set aside.
Heat 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the onion and cook, stirring occasionally, until tender and lightly browned, about 8 minutes. Season with salt and pepper. Transfer to a bowl and wipe out the skillet.
Preheat the broiler. Line a baking sheet with foil. Form the beef into four 1/4-inch-thick oval patties; season with salt and pepper. Transfer to the prepared baking sheet. Broil until cooked through, about 3 minutes. Top with the sauteed onion and cheese and continue broiling until the cheese is just melted, about 30 seconds.
Spread 4 slices bread with the remaining 2 tablespoons dijonnaise; top with the patties and remaining bread. Melt the remaining 1 tablespoon butter in the skillet over medium-high heat. Working in batches, add the sandwiches and cook until golden, 1 1/2 minutes per side, reducing the heat as needed. Serve with the apple slaw.
Photograph by Ryan Dausch
Recipe courtesy of Food Network Magazine