Heat oven to 400° F. Heat the oil in a large skillet over medium heat. Add the onion, garlic, mushrooms, salt, and pepper, and cook until lightly browned, about 5 minutes. Add the tomato paste and cook for 1 minute. Remove from heat and place half the mixture in a food processor. Place the other half of the mixture in a large bowl and set aside.
Add the edamame and parsley to the processor and pulse until the mixture is fine. Transfer to the bowl containing the original mushroom mixture, add the egg and bread crumbs, and mix well. Form into 4 patties and place on a lightly greased sheet pan.
Place the burgers in the oven and bake for 20 minutes. Serve on the buns, topped with sprouts, lettuce, and tomato, if desired.
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