Pre-heat the oven to 350°F.
Heat a sauté pan and caramelize the onion in the oil. Stir in the beets, mushrooms, paprika and cayenne and cook for about 15 minutes, until the mushrooms are soft and most of the moisture has cooked out. Cool to room temperature and season with salt and pepper.
Mix the vegetables with the rice or millet. Transfer the mixture to a food processor and pulse until the vegetables are broken down into small bits, about a minute. Transfer to a bowl and mix in the egg and walnuts.
Oil your hands and form the dough into 6-8 patties.
Place the patties on a greased baking sheet and brush the tops with oil. Bake for 45 minutes, or until the burgers form a light crust. Flip the burgers and cover with the cheese. Bake for another few minutes until the cheese is melted. Serve hot, topped with plenty of dill and your favorite condiments.