In a large, heavy saucepot fitted with a deep-frying thermometer, heat the oil to 350 degrees F. Line one baking sheet with paper towels; line a second baking sheet with parchment; and set a rack over a third baking sheet.
Cut the ends off the onions and peel the skin. Slice the onions crosswise into three rounds, then separate into rings.
Place the dill pickle chips on the paper-towel-lined baking sheet, cover with more paper towels and dry well.
Whisk the eggs in a medium bowl. Add the buttermilk and whisk to combine. In another medium bowl, whisk together the flour, 1 tablespoon salt and 1/2 teaspoon pepper. Place the breadcrumbs in a shallow dish.
Toss the onions rings in the seasoned flour; coat well. Then dip the onion rings in the buttermilk mixture. Dredge the onion rings a second time in the flour and again in the buttermilk (for a double dredge). Toss the onion rings in the breadcrumbs and place on the parchment-lined baking sheet.
Use the same double-dredge process--flour, buttermilk, flour, buttermilk, panko--for the pickle chips.
In batches, fry the onion rings until golden brown, 1 to 1 1/2 minutes. Remove to the rack to rest. Sprinkle immediately with salt.
Fry the pickle chips until golden brown, 1 to 2 minutes. Remove to the rack and sprinkle with salt. Serve immediately.