1. Preheat oven to 400 degrees. Wash beets, wrap individually in foil and place on a baking sheet. Roast until tender, 1 hour. Let cool, peel and coarsely chop. Set aside 4 cups chopped beets and reserve remainder for another use. Reduce oven temperature to 350 degrees.
2. Spread walnuts over a baking sheet. Toast in oven until fragrant, about 5 minutes.
3. In a food processor, pulse toasted walnuts and 4 cups chopped beets until consistency is similar to couscous. Mix in egg, potato starch and bread crumbs. Season with salt and pepper to taste.
4. Shape beet mixture into 8 patties and place on a parchment-paper-lined baking sheet. Bake until slightly firmed, 10 minutes.
5. Heat olive oil in a non-stick pan over medium-high heat. Working in batches and adding more oil as needed, cook patties until a nice crust forms, 2-3 minutes per side.
6. Toast buns. Top bottom buns with beet patties, arugula and tomatoes. Top with remaining buns.