In a large skillet, heat 1 tablespoon oil over medium-high. Add mushrooms; season with salt and pepper. Cook, stirring occasionally, until mushrooms release their liquid and start to brown, 5 to 7 minutes; transfer to a bowl (reserve skillet).
Add onion and remaining tablespoon oil to skillet; season with salt and pepper. Cook, stirring occasionally, until onion starts to brown, 5 to 7 minutes (if bottom of skillet becomes too dry, add a little water and scrape up browned bits). Return mushrooms to skillet; stir in vinegar and set aside.
Heat broiler, with rack set 4 inches from heat. Gently form meat into four 3/4-inch-thick patties; season with salt and pepper. Place on a rimmed baking sheet, and broil 3 minutes.
Remove baking sheet; top burgers with tomato and cheese slices. Broil until cheese is melted and burgers are medium-rare, 1 to 2 minutes more. Serve burgers on toasted bread, with sauteed mushrooms and onion alongside.