For the onion: Trim the stem end of the onion by slicing off 1/2 inch. Peel off and discard any onion skin. Trim the root end of the onion to remove any hair or dirt. Put the onion on a cutting board root-end up. Leaving 1/2 inch uncut around the root end, make 4 vertical cuts evenly spaced around the onion. Continuing this action, make 3 cuts in each quadrant of the onion to create 16 "petals" evenly spaced around the onion root.
Put the onion stem-side up in a medium bowl and sprinkle with some salt and pepper. Mix the buttermilk and hot sauce in a measuring cup and pour over top of the onion. Allow the onion to rest in the buttermilk mixture for 30 minutes.
For the dipping sauce: Meanwhile, whisk together the mayonnaise, horseradish, mustard, lemon juice and Cajun seasoning in a small mixing bowl. Set aside. (See Cook's Note.)
For the breading: Use a mixing bowl that is wide enough to accommodate the onion blossom, and be sure to have a plate that fits over the bowl. Whisk together the flour, cornstarch, dried parsley, Cajun seasoning and ground mustard. Season with 1/2 teaspoon salt and 1/4 teaspoon ground black pepper.
Pour the vegetable oil into a large, heavy-bottomed stockpot. Preheat the oil to 375 degrees F over medium-high heat (the oil will drop in temperature when the onion is added, so adjust the heat as needed to maintain 360 degrees F). Line a baking sheet with paper towels.
Remove the onion from the buttermilk mixture and allow any excess to drip off. Crack the eggs into the buttermilk mixture and whisk to combine.
Place the onion into the seasoned flour, root-end up. Put the plate on top of the bowl to cover the opening, then invert the bowl and plate, holding them together and gently shaking to ensure that the onion is evenly coated in the seasoned flour. Invert again right-side up and remove the plate. Lift the onion out and shake to remove any excess flour.
Place the onion into the egg mixture root-side down, being sure that it is fully submerged. Remove the onion and allow any excess liquid to drip off. Place the onion back into the seasoned flour and repeat the shaking process for a second time to ensure that all petals are evenly breaded.
Carefully lower the breaded onion, root-side down, into oil. Fry until evenly golden brown and crispy, 5 to 6 minutes. Lift the onion out of the hot oil and place on the lined baking sheet. Sprinkle with salt and serve immediately with the dipping sauce.
The dipping sauce can be made in advance and stored in an airtight container in the refrigerator. The longer it sits, the better the flavor gets!