Preheat oven to 375°. Spread beans in an even layer on a large baking sheet. Place mushrooms and onion on another large baking sheet and drizzle with olive oil. Season with salt and pepper and toss to coat, then spread in an even layer. Bake beans until they are dry and the skins are beginning to split, 7 to 9 minutes. Bake mushrooms and onions until tender, 18 to 20 minutes.
Add beans, mushrooms, and onion to the bowl of a food processor, along with oats, avocado, cilantro, garlic, paprika, chili powder, and cumin. Blend until smooth, scraping down sides of food processor as needed. Let mixture rest 10 minutes in refrigerator, then form into 4 patties.
In a large skillet over medium heat, heat enough vegetable oil to coat the bottom of the pan. Add veggie burgers to skillet and cook until outside is crisp and golden, and inside is heated through, about 4 minutes per side.
Assemble burgers with preferred toppings.
