How to Make It
Beat brown sugar, butter, eggs, and vanilla with a heavy-duty stand mixer on medium speed until mixture is blended and smooth.
Process flour, ginger, baking powder, and salt in a food processor until ginger is very finely chopped and mixture is well blended, 1 to 2 minutes. Gradually add flour mixture to sugar mixture, and beat on low speed just until blended after each addition. Shape dough into a disk, and wrap in plastic wrap. Chill 4 to 24 hours.
Preheat oven to 350°F. Shape dough into about 32 (1 1/2-inch) balls. Flatten each dough ball in the palm of your hand, and place 1 caramel half in center of dough. Wrap dough around caramel, and reshape into a ball by rolling in hand, completely covering caramel. Place dough balls 2 inches apart on parchment paper-lined baking sheets. Sprinkle each cookie lightly with Demerara sugar. (Chill assembled balls until ready to bake.)
Bake in preheated oven until flattened and lightly browned on the bottom, 11 to 12 minutes, switching pans top rack to bottom rack halfway through baking. Cool on pans 8 minutes. Serve warm.