Wash the quinoa thoroughly in cold water 7 times. Each time rubbing the grains together firmly to remove the bitter coating. Bring the quinoa to a boil with 2 cups cold water.
Once the water starts to boil, let it go for another 10 to 15 minutes, drain off whatever water remains and cool.
Heat the olive oil in a large skillet over medium heat. Add the onion and garlic and fry until glassy looking. Add the mushrooms and cook until well coated with the onions and just beginning to take on color. Stir in the carrot and parsley. Remove from the heat.
Add this mixture to the quinoa and salt to taste. Add the egg and then the flour in small amounts until you can form small balls that hold together. Leave the balls to rest for at least half an hour in the refrigerator.
Shape and flatten each ball into a patty about 1/4-inch thick. If desired, gently coat each patty in breadcrumbs.
Shallow fry in a pan or place them on a hot oiled baking tray and bake at 350 degrees F.
Cook until lightly browned on each side, about 3 minutes per side. Serve with grilled spring onions and sautéed cherry tomatoes, if desired.