In a large bowl, combine the ground beef, cooked quinoa, Parmesan cheese, egg, garlic powder, onion powder, smoked paprika, salt, and pepper. Mix until just combined, being careful not to overwork the meat.
Form the mixture into four equal patties. Make a slight indentation in the center of each patty to prevent them from puffing up during cooking.
Heat a grill or skillet over medium-high heat. Cook the patties for about 4-5 minutes on each side, or until they reach your desired doneness. In the last minute of cooking, place a slice of cheddar cheese on each patty to melt.
While the burgers are cooking, mix the Greek yogurt and sriracha in a small bowl to create a spicy sauce.
To assemble, toast the whole wheat buns, then layer with arugula, the cooked burger patty with melted cheese, avocado slices, and a generous drizzle of the yogurt sauce. Top with the other half of the bun and serve immediately.