1. In a large skillet, heat canola oil over medium-high heat. Add onion and cook until golden, about 3 to 4 minutes. Stir in mushrooms, garlic, cumin and paprika and cook until mushrooms have released their juices, about 4 to 6 minutes. Remove from heat and set aside to cool slightly.
2. Tear toasted bread into pieces and place in food processor. Process until broken down into crumbs. Pour the crumbs into a medium bowl. Add the mushroom mixture and beans to the food processor; pulse until combined, but not smooth - leave some chunky bits. Scoop mixture into the bowl with the bread crumbs; add the cheese and season with salt and pepper.
3. Mix ingredients together and divide into 4 equal portions. With dampened hands, shape each portion into four thick patties. Lightly brush each side of all patties with canola oil.
4. Heat a grill, grill pan or skillet over medium heat. Place patties on the surface and cook until thoroughly heated through and the outside has crisped, about 5 minutes per side. Serve burgers on toasted English muffins and topped with salsa and sour cream, if desired.