Heat 4 cups vegetable oil in a large pot over medium-high heat to 375°F (190°C). Cut off the top of the onion, peel it, flip it root-side up, and make 16 cuts stopping 1/2 inch from the root. Flip it right-side up and gently spread the petals apart.

Whisk together the flour, paprika, 1 tsp cayenne, garlic powder, onion powder, oregano, salt, and black pepper in a large bowl. Coat the onion in the flour mixture, pressing into every petal. Dip into a bowl of the eggs whisked with the milk, let the excess drip off, then coat in the flour mixture a second time.

Stir together the mayonnaise, horseradish, ketchup, smoked paprika, and 1/4 tsp cayenne in a small bowl. Season with salt, then refrigerate while you fry.

Lower the coated onion petals-side down into the 375°F (190°C) oil. Fry for 5 to 6 minutes, then flip root-side down and fry for another 3 to 4 minutes until deep golden brown. Transfer to a wire rack and drain for 1 to 2 minutes.

Place the onion on a large plate petals-side up, sprinkle with a pinch of salt, and serve immediately with the dipping sauce in the center or alongside.