For the marinade and chicken: In a medium bowl, whisk together the gochujang, brown sugar, vinegar and gochugaru. Transfer 1/4 cup of the mixture to a zip-top bag holding the chicken. Seal the bag and massage the marinade into the chicken. Refrigerate for at least 4 hours or up to overnight. Reserve the remaining mixture to make the gochujang glaze (refrigerate if the chicken is marinating overnight).
In a large Dutch oven, heat the oil until a deep-frying thermometer registers 350 degrees F.
In a pie dish or shallow bowl, whisk together the eggs and buttermilk. In a separate pie dish or shallow bowl, whisk together the cornstarch and self-rising flour. Mix 3 tablespoons of the buttermilk-egg mixture into the flour mixture; lightly whisk to work the buttermilk into the dredge to create little crispy, craggily bits.
Remove the chicken from the marinade and discard the marinade. Dunk the chicken first in the buttermilk-egg mixture, then dredge in the flour mixture to coat thoroughly. Working in batches if necessary, gently drop the chicken into the oil. Cook until the internal temperature of the chicken is 160 degrees F, 8 to 10 minutes. (With the carryover time, it should go up to 165 degrees F.)
For the ginger mayo: Whisk the mayo, ginger, soy sauce and vinegar together.
To serve: In a small saucepan, melt the butter over medium heat. Whisk the butter into the bowl of the reserved marinade until smooth.
Spread each bun top with the ginger mayo. Top with some kimchi.
Toss the chicken in the glaze and place on the bun bottoms. Sprinkle the chicken with sesame seeds. Close the sandwiches.