Grate the beetroot on the coarse side of a box grater, then wrap it in a clean kitchen towel and squeeze hard over the sink until only damp shreds are left. Stir the ground flaxseed with 5 tbsp water in a small bowl and let it sit for 5 minutes until gloopy.
Mash the black beans in a large bowl with a fork or potato masher until about two-thirds are broken and the rest stay whole. Add the squeezed beetroot, brown rice, oats, red onion, garlic, tomato paste, soy sauce, paprika, cumin, salt, pepper and the flax mixture. Mix with your hands for about 1 minute until it sticks together when pinched.
Divide the mix into 6 portions of about 130 g. Press each into a patty about 9 cm / 3.5 inches wide and 2 cm / 3/4 inch thick on a parchment-lined tray. Cover and chill for 30 minutes.
Heat 1 tbsp olive oil in a 30 cm / 12-inch skillet over medium heat for 1 minute. Cook 3 patties for 4 to 5 minutes per side, flipping once, until the crust is dark and crisp and the patty feels firm when pressed. Move them to a plate, add the remaining 1 tbsp oil and cook the other 3 patties the same way.
Toast the buns cut-side down in the same pan over medium heat for 1 to 2 minutes until lightly golden. Spread each bottom with 1 tbsp vegan mayo, then layer lettuce, a patty, tomato and avocado, and close with the top.