In one of the bowls, whisk together the 1 ½ cups of flour with the paprika, garlic powder, salt, and black pepper.
Slowly whisk the beer into the mixture. The goal consistency should be smooth and thick, but still fluid and drippy. (If it looks like cookie dough, it's way too thick. If it looks like pourable pancake batter, it's too thin.) Add more beer if it appears to be necessary. Let the batter rest at room temperature while you prepare the rest of the recipe.
Place the remaining flour in the other bowl.
Peel and slice the onion into ¼-inch thick rings. Keep all but the very smallest rings from the center — it can all be fried up!
Toss the onion rings in the remaining flour to lightly coat them.
Place the oil in the frying pan, with a depth of roughly 3 inches. Heat the oil to 375 degrees fahrenheit, using the candy/oil thermometer to gauge the temperature. It might take time for the oil to heat up. Be patient. Once you reach the temperature, check it occasionally to make sure it remains constant.
When the temperature is correct, take a ring out of the flour and dip it into the batter mixture. Give the ring a little shake and allow the excess batter to drip off. Then carefully place the onion ring into the hot oil. (Don't drop the onion ring in, as that can result in splattering.) Continue to do this, one onion ring at a time, until you have enough rings in the pan without overloading the rings. In my pan, which is roughly a 9" pan, this was about 8 rings.
Fry the first batch of rings roughly 2 to 3 minutes on the first side, then flip and fry the other side for 2 to 3 more minutes, for a total of 4 to 6 minutes for each ring. The goal is less about time and more about color. You are shooting for a light-to-medium golden hue. (See photograph as an example.) So use color over time when determining doneness.
Place the fried rings on a plate lined with paper towels to absorb any excess grease. If desired, you can lightly salt the finished rings (and if you decide to do this, you should do this immediately, while they are still HOT). I did NOT salt the cooked onion rings. I felt like they had enough salt in the batter itself. You can always salt and taste-test one ring first and compare it to an unsalted ring, to see if the rings really need salt or not.