1. Combine the quinoa, eggs, salt, chives, shallots, Parmesan, and garlic in a medium bowl. Stir to blend. Mix in the bread crumbs and stir gently until evenly incorporated. Let sit for 5 minutes. Form the mixture into six evenly sized patties, about 3-4 inches in diameter.
2. 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 3-4 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.