Pulse the mushrooms in batches in a food processor until finely chopped. Set aside.
Heat the oil in a large nonstick skillet over medium-high heat. Add the onions, 1 tablespoon beer and 1/4 teaspoon salt and cook, stirring frequently, until the onions are lightly browned, about 5 minutes. Add the walnuts and peppers and cook, stirring occasionally, until the peppers are crisp-tender, about 3 minutes. Add the mushrooms, 3/4 teaspoon black pepper and chipotle powder and cook, stirring frequently, until the mushrooms are just cooked through, about 5 minutes.
Add the remaining beer, the ketchup, tomato paste and 1/8 teaspoon salt and cook while stirring until the sauce is the consistency that you like, about 2 minutes.
Spoon the mixture onto each bun. Serve with toppings if desired.
Some buns are not vegan so check the label to be sure your hamburger buns are dairy and egg free.