Wrap beets in foil and roast at 450℉. Remove beets from foil. push skins off with fingers. Grate in food processor. Transfer to a bowl and set aside.
Place garbanzo beans, oats, onions, garlic, eggs, basil, smoked paprika, cumin, salt, and cayenne pepper in food processor. Pulse until small chunks of garbanzo beans and oats still remain.
Transfer mixture to a large mixing bowl and blend in the grated beets. Cover bowl with plastic wrap and place in refrigerator so that oats absorb moisture.
Meanwhile, prepare goat cheese spread. Place all ingredients in a bowl and blend together.
Heat 1 tablespoon of coconut oil in a large skillet. Remove beet mixture from refrigerator and form 12 small patties. Cook 4 minutes per side until lightly browned. Add coconut oil to pan for repeated batches.
Place 3 patties in each coconut wrap with sprouts, avocado slices, and chopped lettuce. Top with goat cheese spread.