Bring 2 quarts of water to a boil in a saucepan. Add pork chunks into the water and blanch for 5 minutes. Drain well and place on a large plate lined with a couple layers of paper towels. Set aside.
Heat up a large sauté pan or a wok over medium-low heat for about 5 minutes. Add oil and 2 tablespoons brown sugar. Stir gently until sugar is dissolved in the oil.
Meanwhile, pat the pork dry with additional paper towels. Once the sugar is melted, increase heat to medium and carefully add pork into the pan. (The oil might pop due to excess water from the pork. Don’t stay too close to the stove and be careful.) Brown all sides of the pork. Flip and let it brown undisturbed, about 30-45 seconds on each side.
Once the pork is browned, add dark soy sauce into the pan, stir to combine. Let it cook for 5 minutes.
Now, place regular soy sauce, cooking wine, broth, star anise, green onions, ginger, and garlic into the pan. Depending on the size of your cookware, the cooking liquid should come halfway up the pile of the park. if not, simply add more broth. Bring to a boil, then switch to low heat to simmer, covered, for at least an hour, stirring occasionally.
Uncover, add eggs and gently push it into the cooking liquid, basting for a few times. Cover, and simmer for another 15-20 minutes.
When the pork is fork tender, sprinkle 1 tablespoon brown sugar and increase heat to medium, cook uncovered to let the cooking liquid evaporate and thicken, about 10 minutes.
When ready to serve, transfer eggs to a plate and cut in half. Serve the pork belly with braised eggs over rice.