Preheat oven to 375ºF. Line a baking tray with parchment paper. Brush or spread the oil on the parchment paper and set aside.
Place beets, carrots, onion, mushrooms, and garlic in the bowl of a food processor. Pulse until finely chopped. Empty into a large bowl. Next, place oats, walnuts, chili powder, paprika, cumin, and salt into the processor and pulse several times, leaving some texture to the nuts and oats. Empty into bowl with beet mixture. Finally, place beans and mayonnaise in the processor and pulse to combine, leaving some texture to the beans. Empty into the bowl with the other ingredients and mix everything together.
Using a medium portion scoop or ⅓ measuring cup to scoop mixture onto the prepared baking tray. Use an offset spatula or knife to flatten and create uniform burgers that are 1 1/2 inches thick and about 3 to 4 inches wide. Place tray in oven and bake for about 30 minutes. Remove tray from oven and let burgers sit for about 5 minutes to set. Serve alone or on buns with all of the fixings. Refrigerate leftover cooked burgers for up to 5 days, or freeze between layers of parchment paper in a covered container.