Combine mushrooms, celery, carrot, onion, and beet in a food processor. Blend into a chunky paste.
Heat 1 tablespoon oil in a nonstick skillet over medium heat. Add vegetable paste and fry until the sharp onion taste is gone, about 10 minutes.
Combine cooked rice, flaxseed meal, sunflower seeds, maple syrup, liquid amino acids, fennel seeds, sage, red pepper flakes, nutmeg, black pepper, and thyme in a bowl. Stir until evenly mixed. Add the cooked vegetable paste and stir until sausage mixture is thoroughly combined.
Heat remaining 1 tablespoon oil in a clean nonstick skillet over medium heat. Scoop 1 tablespoon sausage mixture into a 1 1/2-inch patty and place in the hot skillet; repeat with remaining mixture. Brown for 3 minutes; flip and cook until browned on the other side, about 2 minutes more. Add more oil to the pan as needed.