Cut pork into eight 2x3-inch pieces.

Heat oil in a 6-quart Dutch oven over high heat. In 2 batches, sear pork, fat sides first, until light golden, 30 to 60 seconds per side. Transfer to a plate. Pour out all but 1 to 2 tablespoons of the drippings.

Add 1/3 cup sugar to pot; melt, stirring if needed, over medium-high heat. Cook until a light caramel color, about 1 minute. Reduce heat to medium. Return pork to pot, turning pieces to coat. When sugar darkens to orange color, after about 1 minute, add water; stir to loosen any sugar stuck to the bottom.

Add green onions, garlic, sherry, soy sauce, molasses, star anise, and remaining 2 tablespoons sugar. If needed, add up to 1/2 cup additional water to nearly cover pork. Bring to a boil, then reduce heat to medium-low. Simmer, covered, turning occasionally, for about 1 hour and 45 minutes.

Transfer pork to a plate with a slotted spoon; loosely cover. Skim fat and drain solids from cooking liquid. Return 4 cups liquid to pot and bring to a boil; simmer over medium-high heat until reduced to about 1 cup, about 30 minutes.