Bring stock to the boil in a saucepan over high heat. Stir in quinoa; reduce heat to low and cover. Cook until stock is absorbed, about 15 minutes. Remove from heat; let stand for 5 minutes. Fluff with a fork; let cool.
Chop spinach in a food processor; transfer to a bowl. Chop onion, Parmesan cheese and garlic in the food processor; add to the spinach. Mix in cooled quinoa, breadcrumbs, salt, chilli flakes, garlic powder and black pepper. Add egg whites and egg; stir burger mixture to combine.
Shape burger mixture into 4 equal patties. Heat a frying pan over medium-high heat and coat with cooking spray. Cook patties until cooked throughout, 4 to 6 minutes per side. Serve on hamburger buns.
Make sure to flip the patty carefully when cooking to prevent breakage.