Fill a large pot or Dutch oven with enough oil to submerge the onion, with space around the sides for it to "bloom." (You might need up to a gallon, depending on the size of your onion, but don't worry; you can strain the oil and use it again for frying, so it won't go to waste.) Preheat to 400 degrees F.
Combine the milk, eggs, garlic, 1/2 teaspoon salt and a generous amount of pepper in a medium bowl. Whisk to combine.
Combine the flour, Parmesan, paprika, garlic powder, 1 teaspoon salt and a generous amount of pepper in a large bowl. Toss with a fork to combine.
Cut off the top of the onion and place cut-side down. Think of the onion as the face of a clock, and cut slits at noon, 3, 6 and 9, cutting almost but not quite to the center of the onion. You'll now have 4 quadrants. Cut 3 more slits in each quadrant, once again cutting almost but not quite to the center of the onion. Flip the onion and gently separate the "petals" with your fingers.
Put the onion in the flour mixture and sprinkle it over and around to coat, gently separating the petals as needed to completely coat it.
Put the onion in the egg wash and coat evenly in the egg, opening the petals again as needed. Let the excess egg wash drip back in the bowl.
Put the onion in the flour once more and sprinkle it over and around to coat, making sure to keep the petals as separate as you can.
Use a large spider to lower the onion, cut-side up, into the hot oil. The temperature of the oil will immediately begin to drop. Adjust the heat so the oil stays around 350 to 360 degrees F for the remainder of frying. With a fork, separate the petals a bit more, if needed. Fry the onion until the tips are golden, 3 to 4 minutes. Use the spider to gently flip the onion and continue to fry until the entire onion is crisp, deep golden and cooked through, 3 to 4 minutes more. Drain on a paper towel-lined plate.
Transfer the onion while still hot to a serving plate. Season with salt and pepper. Shower with chopped parsley and grated Parmesan. Serve with mayo for dunking.