- Thinly slice the mushrooms.
- Cut a small corner from the burger packaging and drain off any excess liquid. Transfer to a plate; pat dry with a paper towel. Season with salt and pepper. (If using Impossible Burger, season with salt and pepper; using wet hands, form into two [four] ½-inch-thick patties.)
In a large frying pan over medium-high heat, warm 1 tablespoon oil until hot but not smoking. Add the patties and cook, turning once, until browned, 2 to 4 minutes per side for medium-rare. Transfer to a plate. Do not clean the pan.
In the same pan, if dry, add 2 to 3 teaspoons oil. Warm over medium heat until hot but not smoking. Stir in the mushrooms, season with salt and pepper, and cook, scraping up any browned bits from the bottom of the pan, until starting to brown and soften, 2 to 3 minutes. Stir in the caramelized onions and cook until heated through, about 1 minute.
Push the mushrooms and onions to one side of the pan and return the burgers to the other side. Spoon about one-third of the mushrooms and onions over each burger, then top with the Swiss cheese. Remove from the heat and cover until the cheese melts, 1 to 2 minutes.
While the burgers are cooking, prepare the salad.