Mince the onion and place it in a microwave-safe container. Cover loosely with plastic wrap and microwave at 600W for 30 seconds. Let it cool. Chop the kimchi into small pieces.

In a bowl, mix breadcrumbs and milk to moisten.

Add ground chicken, minced onion, chopped kimchi, and salt and pepper to the breadcrumb mixture. Mix until sticky, then divide into two portions. Shape each into a patty, pressing out air.

Heat vegetable oil in a frying pan. Add the patties, pressing the center to create an indentation. Cook over medium heat until browned, then flip. Cover and cook on low heat for 7–8 minutes.

Uncover, place a slice of melting cheese on each patty, cover again, and cook on low heat until the cheese melts. Remove the patties from the pan.

In the same pan, add sake, mirin, and soy sauce. Bring to a boil and stir over medium heat until the sauce thickens.