Cut the pork belly and pork shoulder into 2-inch chunks and place them in a pot with cold water, ginger slices, and a pinch of salt. Bring to a boil, skim off any impurities, then drain and rinse the pork thoroughly under cold water to keep the braising liquid clear.
Transfer the pork to a bowl and season with fish sauce, chicken bouillon, salt, sugar, freshly cracked black pepper, and a squeeze of lemon or lime juice. Mix well until evenly coated and let marinate for about 30 minutes.
While the pork marinates, prepare the eggs by tapping the rounded end of each egg with a spoon until you hear a crack to break the membrane. Boil the eggs with a splash of vinegar until just set, about 7–8 minutes for chicken eggs or about 4 minutes for quail eggs, then cool, peel, and set aside.
Place a heavy pot or braiser over medium-low heat and add the sugar with a small splash of water. Allow the sugar to melt and caramelize without stirring, watching closely until it reaches a deep golden amber, then add a splash of neutral oil to loosen the caramel.
Carefully add the marinated pork to the pot and gently toss to coat all pieces evenly in the caramel.
Pour in enough coconut water to fully submerge the pork, bring to a gentle simmer, and skim off any foam that rises to the surface. Add the whole garlic cloves and shallots, partially cover the pot, and simmer gently without boiling rapidly for 60–75 minutes, until the pork is about 75% tender.
Remove the garlic and shallots, add the eggs to the pot, and continue simmering gently until the pork is fully tender and glossy and the eggs have absorbed some of the braising liquid.
Taste and adjust seasoning if needed, then serve hot with steamed white rice and dưa giá.