For pickled onions, place onion slices in a medium bowl. Pour vinegar and warm water over onion. Stir in sugar and season with a pinch of salt; let stand for about 30 minutes or until slightly pickled. Drain and set aside.
Meanwhile, in a medium saucepan, cook the quinoa in boiling water until just tender, about 10 minutes (or according to your package instructions). Drain and spread on a baking sheet to cool.
In a bowl, combine eggs, carrot, cheese, chives, garlic, salt, and pepper. Mix in the cooled quinoa and bread crumbs. Let stand for 10 minutes.
Form quinoa into ½-inch-thick patties; you should get about 12 slider-sized patties or 6 larger ones. Place on a lightly oiled platter.
In a large nonstick skillet, heat 1 tablespoon of the oil. Add half of the patties and cook until golden brown, about 3 to 5 minutes. Flip and cook for 3 to 5 minutes more or until crisp, adding more oil as needed. Repeat with the remaining patties and oil.
To assemble, spread 1 to 2 tablespoons of goat cheese onto the bottom of each burger bun, then top with burger patty, tomato slices, arugula, and pickled onions.
