Place the beets, ½ cup oats, beans, walnuts, celery, onion, and garlic into a food processor. Process lightly just until the mixture comes together but still has some medium pieces and texture. Don't overprocess or it will become pasty.
Transfer the mixture to a large mixing bowl. Add ½ cup oats, breadcrumbs, mustard, flaxseed, thyme, salt, and pepper and mix well with your hands. Taste and adjust seasonings if desired. If it seems overly soft, you can add up to 2 more tablespoons of breadcrumbs.
Place the bowl in the refrigerator and let it chill for about 30 minutes.
Coat a pan, ideally cast iron, with a thin layer of olive oil or spray with non-stick cooking spray. Preheat the pan to medium high.
While the pan heats up, shape the mixture into 8 patties with your hands.
Carefully place patties into the hot pan. Depending on the size of your pan, you may have to cook them in more than one batch.
Cook the patties for about 8 minutes per side until the outside is crispy and charred and the inside is heated through. Flip them once when the first side looks done. If needed, add a small amount of additional oil to the pan when you flip them.
Serve immediately on a burger bun with your favorite toppings, or for a firmer burger, let cool and reheat before serving.