Pulse the mushrooms in a food processor until finely chopped. Transfer to a medium bowl; add the beef, garlic, soy sauce, 1/4 teaspoon salt and a few grinds of pepper and mix with your hands until combined. Form into four 1/2-inch-thick patties.
Heat 1 tablespoon vegetable oil in a large nonstick skillet over medium-high heat. Add the patties, season with salt and cook until browned, about 3 minutes per side for medium rare. Top each burger with a slice of Swiss cheese, cover and cook until the cheese melts, about 1 more minute. Remove from the heat and uncover.
Meanwhile, put the edamame in a microwave-safe bowl; cover with plastic wrap and pierce a few times with a knife to vent. Microwave until the edamame is bright green and tender, about 5 minutes. Uncover and toss with the remaining 2 teaspoons vegetable oil; season with salt.
Toast the English muffins, then sandwich with the burgers and toppings. Serve with the edamame.