Preheat the oven to 350 degrees F.
Line two baking sheets with aluminum foil and spray with nonstick spray. Spread the beans on one sheet in a single layer. Spread the chopped broccoli on the other sheet in a single layer. Sprinkle the broccoli with salt and pepper. Roast the beans and broccoli for 12 to 15 minutes, until the beans split open and the broccoli cooks a bit.
While the beans and broccoli are roasting, heat a skillet over medium low heat. Add the olive oil, diced onion and garlic and cook until softened, about 5 to 6 minutes.
At this time, you can also pulse the cashews and oats in your food processor until they are finely chopped. Don’t wash the food processor because we will be using it again!
In a large bowl, stir together the mayo, egg yolk, smoked paprika and honey. Stir in the sautéed onion mixture, quinoa, the beans and the broccoli. Add in the oats and the cashews. Use a potato masher or fork to smash the mixture together until it is somewhat combined. Take 1/3 of the mixture and place it back in the food processor. Pulse it a few times until it’s all very finely chopped and combined, then transfer it back to the large bowl. Stir in the cheddar cheese, salt and pepper. Mix everything together (I suggest using your hands – it will be sticky!) until combined.
At this point, form the mixture into patties or burgers – whatever the size of your bun is! You can go with 4 or 6 patties - or even smaller for slider size. You can stick the patties in the fridge until it’s time to use them, meaning that you can make them ahead of time. You don’t have to refrigerate them before cooking, but sometimes I find that refrigerating them for 20 minutes can be helpful.
When you’re ready to cook, heat a large nonstick skillet over medium heat. Add a high heat oil – like canola, grapeseed or coconut. Once the oil is hot, add the patties and cook them for 5 to 6 minutes on each side. Be gentle when flipping to ensure they don’t fall apart! A minute or so before removing them from the skillet, add a slice of cheddar on top so it melts.
Serve on buns with the spicy mayo and tomato.
As a note, these cannot be grilled on an outdoor grill (like most homemade veggie burgers can’t!) - they will fall apart. You can, however, place a cast iron skillet on the grill and cook them outdoors that way.