Whisk miso, kosher salt, honey, and 1/2 cup water in a medium bowl until smooth. Gradually whisk in another 1/2 cup water; add pork belly. Cover and chill 3 days.
Preheat oven to 325°F. Blend sugar, lime juice, galangal, and 1 1/2 cups water in a blender until smooth. Remove pork from brine and rinse off; discard brine. Place pork and galangal syrup in a large heavy pot. Cover pot, transfer to oven, and braise pork until tender and nearly falling apart, 3–3 1/2 hours.