Sauce Combine mayonnaise, sour cream, ketchup, horseradish, paprika, salt, garlic powder, oregano, black pepper, and cayenne in a bowl, cover, and refrigerate.
Blooming Onion Trim ½ inch off the top of the sweet onion, then peel away the outer layer. Position the onion cut-side down, then starting about ½ inch from the root, slice downward through to the board to form four evenly spaced cuts. Flip the onion over, rest for 5 minutes to allow the petals to loosen, then gently separate the layers with your fingers. Create three extra cuts between each section, producing 16 segments. In a large bowl, whisk flour with paprika, cayenne, cumin, thyme, and oregano. In a separate medium bowl, beat the eggs with milk and 1 cup water. Set the onion in a clean bowl cut-side up, pour the flour blend over, and use a spoon to work the mixture between the petals. Flip the onion over and shake off the excess flour, keeping the remaining flour for a second coating. Dip the onion into the egg mixture, letting the excess drip off, then dredge once more in the reserved flour mixture. Place the coated onion in the refrigerator. Preheat the air fryer to 350°F for 10 minutes. Remove the onion, spray generously with cooking oil, and position it cut-side up in the fryer basket using an aluminum foil sling. Cook for 25 minutes, misting with oil 2–3 times during cooking. Lift out the onion with the foil sling, season with salt and pepper, and serve warm alongside the prepared sauce.