Add butter to a medium pan over medium heat, and let it melt. Add the kimchi and cook until softened, about 5 minutes. Add the brown sugar and mirin, and mix until sugar dissolves. Cover and let it simmer for 5-8 minutes, or until cabbage is throughly cooked.
Transfer to a food processor and blend.
Transfer back to the pan and cook further, stirring occasionally until jammy (about 5-8 minutes).
Transfer to a bowl and cool. Serve or store in airtight container in the refrigerator for up to 2 weeks.
