Peel the onions, slice them 1/2 to 3/4-inch thick, and separate them into rings.
Combine the buttermilk, 1 1/2 teaspoons salt, 1 teaspoon pepper in a medium bowl. Add the onion rings, toss well, and allow to marinate for at least 15 minutes. (The onion rings can sit in the buttermilk for a few hours.)
In a separate bowl, combine the flour, cornmeal, 1 teaspoon salt, and 1/2 teaspoon pepper. Set aside.
When you're ready to fry the onion rings, preheat the oven to 200 degrees F. Line a baking sheet with paper towels.
Heat the oil to 350 degrees F in a large pot or Dutch oven. (A candy thermometer attached to the side of the pot will help you maintain the proper temperature.).
Working in batches, lift some onions out of the buttermilk and dredge them in the flour mixture. Drop into the hot oil and fry for 2 minutes, until golden brown, turning them once with tongs. Don't crowd them!
Place the finished onion rings on the baking sheet, sprinkle liberally with salt, and keep them warm in the oven while you fry the next batch. Continue frying the onion rings and placing them in the warm oven until all the onions are fried. They will remain crisp in the oven for up to 30 minutes. Serve hot.