Heat the oil in a large ovenproof pot over medium heat until shimmering. Sear the pork skin-side down until it starts to turn golden brown, 5 to 6 minutes. Remove from the pot and set aside. Add the onion, garlic, ginger, cloves and star anise and cook, stirring occasionally, until the onions start to soften, 4 to 5 minutes.
Stir in the brown sugar and cook, stirring frequently, until melted, 1 to 2 minutes. Pour in the soy sauce and vinegar and stir to combine. Add the pork back to the pot skin-side down and cover with water (about 10 cups). Bring the mixture to a boil over medium-high heat. Cover and transfer the pot to the oven. Cook until the pork is tender and easily shredded with a fork, 3 1/2 to 4 hours.
Meanwhile, place the eggs in a medium pot and cover by 2 inches with cold water. Bring the water to a boil over medium-high heat. Once boiling, immediately turn off the heat and cover the pot. Let the eggs sit in the hot water for 3 minutes. Drain the water and submerge the eggs in a bowl of ice water to stop the cooking. Peel the eggs and cut 4 to 5 small slits into the whites of each egg. Refrigerate the eggs until needed.
Remove the pork from the pot and set the pot over high heat. Bring the liquid left in the pot to a boil and reduce by half or until thickened, about 30 minutes (you should have 4 cups of liquid left). Add the eggs to the pot and cook for an additional 5 to 6 minutes so the sweet and salty sauce gets into the yolks of the eggs. Strain the sauce into a bowl or gravy strainer and skim off the fat. Slice the pork and spoon the sauce over the top. Serve with the eggs on the side.