Preheat the oven to 200 degrees F. Set a cooling rack over a baking sheet.
Separate the onions into rings and put in a large bowl. Toss with 1/2 cup of the flour. Arrange the floured rings on the cooling rack; let stand for 15 minutes.
Fill a large, wide pot with 2 inches of vegetable oil and heat to 365 degrees F over medium-high heat. In a large bowl, whisk the remaining 2 cups flour with the cornstarch, paprika, cayenne, lime zest and 1 1/2 teaspoons kosher salt. Add the milk, club soda and egg and whisk to combine.
Working in 3 or 4 batches, add a small amount of the onions to the batter and then immediately place into the hot oil. Fry until crisp and golden, turning once, 2 to 3 minutes. Allow the oil to come back to temperature between batches.
Season the onion rings with salt. Serve immediately with your condiment of choice, or transfer to a cooling rack and keep warm in the oven until ready to eat.