Combine cooked quinoa, beans, egg, bread crumbs, onion, bell pepper, Cilantro Lime Aioli or Chipotle Aioli, garlic, salt, cumin, and pepper in a food processor. Pulse several times to combine. Form into patties. Allow to sit for 10 minutes before frying.
Heat oil in a large skillet over medium-low heat. Cook burgers approximately 6 minutes per side. Gently turn burgers and continue to cook until golden brown on both sides.
Place burgers on the bottom of each bun. Top with aioli, lettuce, tomato and avocado. Serve immediately.