Fragrant lemongrass and galangal add brightness and balance to the savory-sweet sauce that coats tender pork belly. Searing the pork in batches ensures the meat browns while the fat renders.
Heat oil in a medium Dutch oven over medium-high. Add half of the pork belly pieces; cook until lightly browned on 2 sides, about 5 minutes per side. Transfer browned pork to a bowl. Drain and discard drippings. Repeat with remaining pork; drain and discard drippings. Stir together quail eggs and 2 tablespoons soy sauce in a small bowl; set aside.
Return browned pork to Dutch oven. Reduce heat to medium; add brown sugar, and cook, stirring constantly, until dissolved, about 10 seconds. Add salt and remaining 2 tablespoons soy sauce; cook, stirring often, until pork is glazed in sauce mixture and darkened, about 5 minutes. Drain drippings, holding pork against side of Dutch oven to keep in place while pouring. Discard drippings.
Add enough water to just cover pork in Dutch oven (about 4 cups); bring to a boil over high. Reduce heat to low; stir in lemongrass and galangal. Partially cover; simmer, stirring occasionally, until pork is tender, 1 hour to 1 hour and 15 minutes, stirring in egg mixture during final 5 minutes of cook time and adding additional water to keep pork just covered in liquid.
Remove from heat. Remove and discard lemongrass and galangal, if desired. Let stand 5 minutes; skim and discard fat from surface. Stir in Chinese five spice, if using. Taste mixture; if desired, add additional soy sauce or brown sugar to achieve desired savory-sweet balance. Serve with cooked rice; drizzle with sauce from Dutch oven.