In a colander toss eggplant with salt and let sit 30 minutes. Rinse eggplant well and pat dry.
Place flour in a wide, shallow bowl. In a second wide, shallow bowl, add 1 cup mayo. In a third, combine panko and sesame seeds. Dredge eggplant slices in flour, then coat lightly in mayo, and dredge in panko mixture. Set on a sheet pan.
Heat oil in a large nonstick skillet over medium heat. When hot, fry eggplant in batches, turning once, until golden brown on both sides. Do not crowd the pan. If needed, wipe out the skillet, add fresh oil, and continue frying.
Spread buns with remaining 1⁄4 cup mayo and top with fried eggplant slices. Drizzle eggplant with Kalbi sauce and top with cabbage and kimchi.