Wash and dry the fresh produce. Halve, peel, and thinly slice the red onion(s). Grate the grand cru on the large side of a box grater. Halve the buns. Roughly chop the cipolline onions. Quarter, core, and thinly slice the apple(s). In a large bowl, whisk together the vinegar, honey, and a drizzle of olive oil; season with salt and pepper.

In a medium nonstick pan (or large for 4 servings), heat a drizzle of olive oil on medium-high until hot. Add the sliced red onion(s); season with salt and pepper. Cook, stirring occasionally, 6 to 8 minutes, or until lightly browned and softened. Transfer to a bowl; cover with foil to keep warm. Wipe out the pan.

In the same pan, heat a drizzle of olive oil on medium-high until hot. Add the patties. Cook 4 to 6 minutes per side, or until browned and heated through. Transfer to a work surface.

Meanwhile, in a small pot, combine the béchamel and mushroom duxelles. Cook on medium, whisking constantly, 2 to 3 minutes, or until the sauce starts to bubble. Add the grated grand cru. Season with salt and pepper. Cook, whisking constantly, 2 to 3 minutes, or until the cheese is melted and combined. Turn off the heat.

Working in batches if necessary, add the halved buns, cut side down, to the same pan. Toast on medium- high 30 seconds to 1 minute, or until lightly browned. Transfer to a work surface.

To the bowl of dressing, add the arugula, chopped cipolline onions, sliced apple(s), and goat cheese (crumbling before adding). Season with salt and pepper; toss to combine. Assemble the burgers using the toasted buns, dijonnaise, cooked patties, cheese sauce, and as much of the cooked onion(s) as you'd like. Serve the burgers with the salad on the side. Enjoy!