Place onion rounds, 2 cups of the beer, cider vinegar, ½ teaspoon salt, and ½ teaspoon pepper in zipper-lock bag; refrigerate 20 minutes or up to 2 hours.

Heat oil in large Dutch oven over medium-high heat to 350°F.

While oil is heating, combine flour, cornstarch, baking powder, ½ teaspoon salt, and ¼ teaspoon pepper in large bowl. Slowly whisk in ¾ cup beer until just combined (some lumps will remain). Whisk in remaining beer as needed, 1 tablespoon at a time, until batter is thick enough that it falls from the whisk in a steady stream and leaves a faint trail across the surface of the batter.

Pour off liquid from onions. Pat onion rounds dry with paper towels and separate into rings. Transfer one-third portion of rings to batter. One at a time, carefully transfer battered rings to the oil. Fry until rings are golden brown and crisp, about 6-7 minutes, flipping halfway through frying. 

Drain rings on wire rack set on baking sheet, and season with salt and pepper. Return oil to 350°F and repeat with remaining onion rings and batter. (If planning on keeping for a period of time, preheat oven to 200°F, and hold rings in there until needed)