Cut off the top of the onion and peel. Place cut side down and make vertical cuts around the onion, stopping 1/2 inch from the bottom, creating 12-16 evenly spaced petals. Soak in ice water for 10 minutes, then drain and pat dry.
In a shallow bowl, combine flour, paprika, garlic powder, cayenne, oregano, cumin, black pepper, and salt. In another bowl, whisk together eggs and milk.
Dip the onion in the seasoned flour, coating thoroughly and getting between all the petals. Dip in egg mixture, then coat again with seasoned flour, making sure every petal is well covered.
Heat 4-5 inches of oil in a large Dutch oven or deep pot to 350°F. Carefully place the onion cut-side down in the hot oil.
Fry for 3 minutes, then carefully flip using tongs or a spider strainer. Fry for an additional 6-7 minutes until deep golden brown and crispy.
Remove from oil and drain on paper towels. Season with additional salt immediately.
Make the dipping sauce by whisking together mayonnaise, ketchup, horseradish, and hot sauce in a small bowl.
Serve the cactus blossom immediately while hot with the dipping sauce on the side.