Remove the papery red onion layers and trim the ends. Then slice crosswise into ½-inch-thick rings. Pat the Wagyu beef patties dry with paper towels. Season generously with salt and pepper.

Separate the brioche buns and lightly brush the cut sides with cooking oil (or mayo for extra browning). Heat a nonstick saute pan over medium-high heat. When the pan is hot, carefully add the buns cut-side down and cook for 90 seconds, or until lightly toasted. Transfer to serving plates.

Add 1-2 tbsp. of cooking oil to the saute pan. When the oil is hot, carefully add the Wagyu patties and cook for 3 minutes, without turning, to form a browned crust. Reduce the heat to medium. Flip the patties and move them to one side of the pan. Carefully add the onion slices to the pan across from the patties. Cook for 4 minutes, or until the onions begin to brown, stirring occasionally.

Pour the balsamic vinegar over the onion slices and stir to loosen any tasty bits stuck to the bottom. Cook for 1 minute, or until the vinegar has reduced slightly and glazed the onion. Spoon the glazed onions on top of the burgers, then top with the Swiss cheese. Cover the pan and cook for 1-2 minutes, then turn off the heat. Leave covered until ready to assemble.

Spread the mayo across the cut sides of the brioche buns. If desired, slice the pickles and place them onto the bottom buns, or keep them whole to enjoy on the side. Place the burgers onto the bottom buns, add a pinch of arugula, then complete with the top buns. Serve the baby carrots and ranch dressing on the side.