Place the sugar in a large skillet over medium heat. Cook without stirring until it melts and turns a deep amber color, about 4 minutes.
Add the pork shoulder chunks to the caramel and stir to coat. Cook for 5 minutes, turning the pieces so they brown lightly on all sides.
Stir in the garlic and shallots. Cook for 2 minutes until fragrant.
Transfer everything to a slow cooker. Add the fish sauce, water, star anise, and black pepper. Stir once.
Cover and cook on low for 7 hours.
Add the peeled hard-boiled eggs during the last 2 hours so they warm through and absorb flavor.
Taste the sauce and adjust with a splash more fish sauce if needed. Remove the star anise.
Ladle the stew into bowls and top with sliced green onions.