In a medium bowl, combine the quinoa, eggs, salt, chives, shallots, Parmesan, and garlic. Stir to blend. Mix in the bread crumbs and stir gently until evenly incorporated. Form the mixture into six evenly sized patties, about 3-4 inches in diameter.
Heat a bit of olive oil in a large sauté pan set over medium-high heat and swirl to coat. Add the patties to the pan so that they are not touching. Let cook about 4 minutes or until the first side is lightly browned. Carefully flip the patties with a spatula and let the second side cook until golden brown, about 4 minutes more. Repeat with the remaining patties, using additional olive oil if needed. Top each patty with a bit of shaved Parmesan, a poached egg, and season with salt and pepper to taste. Serve immediately.