Combine all of the dip ingredients in a bowl, cover and refrigerate.
Slice the onion (see Cook's Note). Whisk the flour, cayenne, paprika, thyme, oregano, cumin and 1/2 teaspoon black pepper in a bowl. In a small deep bowl, whisk the eggs, milk and 1 cup water.
Place the onion in a separate bowl, cut-side up, and pour all of the flour mixture on top. Cover the bowl with a plate, then shake back and forth to distribute the flour. Check to make sure the onion is fully coated, especially between the "petals." Lift the onion by the core, turn over and pat off the excess flour; reserve the bowl of flour.
Using a slotted spoon, fully submerge the onion in the egg mixture (spoon on top, if necessary). Remove and let the excess egg drip off, then repeat the flouring process. Refrigerate the onion while you heat the oil.
Heat the oil in a large deep pot over medium-high heat until a deep-fry thermometer registers 400 degrees. Pat off excess flour from the onion. Using a wire skimmer, carefully lower the onion into the oil, cut-side down. Adjust the heat so the oil temperature stays close to 350 degrees. Fry about 3 minutes, then turn the onion over and cook until golden, about 3 more minutes; drain on paper towels. Season with salt and serve with the dip.