In a large bowl, combine ground beef, cooked quinoa, chopped onion, breadcrumbs, egg, garlic powder, smoked paprika, salt, and pepper. Mix until well combined.
Form the mixture into 4 equal patties, making a slight indentation in the center of each patty to ensure even cooking.
Preheat a grill or skillet over medium-high heat. Cook the patties for about 5-6 minutes on each side, or until they reach your desired level of doneness. In the last minute of cooking, place a slice of cheddar cheese on each patty to melt.
While the patties are cooking, prepare the sauce by mixing mayonnaise, Dijon mustard, and lemon juice in a small bowl.
Assemble the burgers by spreading the sauce on the bottom half of each bun, adding lettuce, the cooked patty with melted cheese, avocado slices, and tomato slices. Top with the other half of the bun and serve immediately.