Pour the beer into a medium bowl and stir in the vodka. Slowly whisk in 1 teaspoon paprika and 1 cup flour. The batter should be fairly thick (to coat the onion rings before frying) but also easy to stir. Set aside in a warm place.
Gather the onion rings in a medium bowl and use a small strainer to sprinkle it with remaining 2 tablespoons paprika. Toss to blend.
Prepare a baking sheet fitted with a kitchen towel to drain the onion rings as they come out of the oil. Stir the batter slightly. To test the consistency of the batter and the temperature of the oil, put an onion ring on a slotted spoon and submerge it completely in the batter. It should coat the piece lightly. If the batter is too thick, add a little water to thin it out. If the batter is too thin, sift and whisk in a little more flour.
Dip the onion ring in the batter, drain the excess batter, and use a slotted spoon or a pair of kitchen tongs to lower the onion ring into the oil. It should bubble slightly and the onion should gradually rise to the top. Fry until light brown, 2 to 3 minutes, and transfer them to the kitchen towel. Fry the onions in small batches. Season with salt and serve immediately.