Preheat oven to 420˚F. Line two baking sheets with parchment or Silpats.
In a wide, shallow dish, combine the quinoa flour, arrowroot powder, paprika, cayenne, and salt. In a second wide, shallow dish, whisk together the buttermilk and eggs.
Place the quinoa and quinoa flakes in a blender or food processor, and pulse until crumbly. Transfer to a third shallow dish.
Dredge the onion rings into the quinoa flour-arrowroot mixture, then dip into the buttermilk mixture. Gently shake off excess batter, then dredge in quinoa-quinoa flakes mixture. Set on the prepared baking sheets in a single layer. Generously spray onion rings with olive oil. Bake 20-25 minutes, flipping midway through cooking and spraying again with olive oil, until golden brown.
In a small bowl, whisk together the Greek yogurt or vegan sour cream, ketchup, horseradish, chives, remaining ½ teaspoon of smoked paprika, and remaining ¼ teaspoon of cayenne. Stir in a pinch of salt and pepper.
Serve the onion rings hot out of the oven, with the dipping sauce on the side.