Slice eggplants crosswise a scant ½-inch thick. Sprinkle both sides generously with salt and spread in a single layer on a rack over a rimmed baking dish; allow to drain for 30 minutes. Rinse thoroughly, place on a clean baking sheet, and sprinkle with hot pepper vinegar, turning several times to coat. Soak for 15 minutes.

Pour cooking oil to a depth of 2 inches in a deep skillet and place over medium-high heat. Bring to 350F. While the oil heats, combine the egg, buttermilk, pepper, and salt in a large shallow bowl. Combine the flour and panko in another large shallow bowl. Dip the eggplant slices in the egg mixture, then the flour and panko mixture, shaking off excess after each dip. Working in batches, fry the eggplant slices, turning once, until golden and crispy on both sides, about 5 minutes total. Transfer to a baking rack set over paper towels to drain; sprinkle with salt and pepper to taste.