Cut top off onion; peel. Place cut-side down; make vertical cuts, stopping 1/2 inch from bottom, to form 12-16 evenly spaced petals. Soak in ice water 10 minutes; drain and pat dry.
Combine flour, paprika, garlic powder, cayenne, oregano, cumin, pepper, and salt in a shallow bowl. Whisk eggs and milk in another.
Coat onion thoroughly in seasoned flour, including between petals. Dip in egg, then flour again, coating every petal well.
Heat 4-5 inches oil in a large Dutch oven or deep pot to 350°F. Carefully add onion cut-side down.
Fry 3 minutes; carefully flip with tongs or spider strainer. Fry another 6-7 minutes until deep golden brown and crispy.
Remove; drain on paper towels and immediately season with additional salt.
Whisk mayonnaise, ketchup, horseradish, and hot sauce in a small bowl.
Serve onion immediately, hot, with sauce alongside.