Begin by rinsing the black sesame seeds thoroughly several times to eliminate any dust or unwanted particles. Once clean, place them in a dry pan over low heat and gently toast until they are completely dried and aromatic, being careful not to burn them.
Transfer the toasted seeds to a food chopper or processor and pulse until they reach a fine, powdery consistency. Add the sugar to the ground sesame and blend again until evenly combined. Drop in the unsalted butter, then process the mixture until it transforms into a cohesive, fragrant black sesame paste.
Scoop the paste into a container and place it in the freezer for about 5–10 minutes, just until it firms up slightly. This makes it easier to handle and shape. Once slightly hardened, portion and roll the paste into small balls about ½ to ¾ inch wide. Arrange them on a tray and freeze overnight. You can store any extras for later use.
In a mixing bowl, combine glutinous rice flour with warm water. Stir and knead the mixture until it forms a pliable, smooth dough. If the dough feels dry or cracks during shaping, add water a little at a time until it becomes soft but not sticky. For a larger batch, the dough can be doubled to accommodate all the black sesame filling.
Divide the dough into equal small pieces. Flatten each one into a disc, keeping the center slightly thicker than the edges to help encase the filling without tearing. Place a frozen sesame ball in the center, wrap the dough around it completely, and roll gently between your palms until you form a smooth, sealed ball. If the dough starts to dry or crack, wet your hands slightly to help seal and smooth the surface.
Bring 6 cups of water to a rolling boil in a pot. Once boiling, reduce the heat to medium-low and gently lower the tang yuan into the water. Allow them to cook undisturbed until they float to the surface, about 5 to 7 minutes. Floating is the sign they’re cooked through and ready.
While the tang yuan cook, prepare the sweet ginger syrup. In a separate pot, combine 2 cups of water, brown sugar, and the ginger strips. Bring to a boil and simmer for 5 to 10 minutes, allowing the ginger to infuse its warmth and spice into the syrup.
Once the tang yuan are done, carefully drain them and transfer into serving bowls. Pour the hot ginger sugar syrup over the dumplings and serve warm.