Bring 2 cups of water to a boil in a medium sauce pan. Add the quinoa, stir, and return to a boil.
Cover and simmer for about 12 minutes. Remove from heat and allow to steam, with lid on, for 3 minutes. Fluff with a fork.
In a medium bowl, combine the cooked quinoa with the carrot, onion, salt, and pepper. Stir in the eggs, bread crumbs, and cheeses until well blended. Form the mixture into balls and then flatten into 1-inch thick patties.
Heat 1 tablespoon of oil in a large skillet over medium heat. Add 4 of the patties and cook 6-8 minutes or until golden brown. Flip and cook an additional 2-3 minutes, until browned. Remove to a plate lined with paper towel to absorb excess oil. Repeat with remaining oil and patties.
While patties cook, combine the mayonnaise, yogurt, salt and pepper in a small bowl. Serve hot quinoa patties with remoulade.