Rinse the quinoa in a large fine mesh strainer with cold water, then transfer to a small saucepan. Cover with 2 cups cold water, bring to a boil, and then reduce to a low simmer and cook for 15 to 20 minutes. Remove from the heat, fluff with a fork and let cool before mixing into the lamb.
Mix together the quinoa, lamb, cumin, cayenne, garlic, egg, onions and some salt in a large bowl until the mixture comes together and is smooth. Form the mixture into 4 balls, flatten down into patties and place on baking sheets.
Heat a large nonstick saute pan over high heat for 1 minute, and then add the oil. Cook the patties until browned, 3 minutes per side, and then cover and cook over low heat for 10 to 12 minutes.
Mix together the yogurt and mint in a separate bowl and season with salt.
To plate the dish, place the lamb burger over your favorite salad or I recommend a fresh tomato puree. Top each with a dollop of the mint yogurt and serve.