Prepare the Cookies: Beat unsalted butter with a heavy-duty electric stand mixer on medium speed until creamy. Gradually add granulated sugar and light brown sugar, beating until light and fluffy, about 2 minutes. Add fresh ginger, sorghum syrup, and vanilla extract; beat until combined. Stir together flour, salt, and ground cardamom in a small bowl. Gradually add flour mixture to butter mixture, beating on low speed until well blended.

Turn dough out onto a clean work surface. Divide dough in half, and pat each half into a 1⁄2-inch-thick disk. Wrap each dough disk in plastic wrap; chill 2 hours.

Preheat oven to 300°F. Unwrap 1 dough disk, and place on a lightly floured surface. Roll dough to about 3⁄16-inch thickness (between 1⁄8 inch and 1⁄4 inch thick). Using a 3 1⁄2-inch light bulb-shaped cutter, cut shapes from dough, rerolling scraps once. Using a straw or metal piping tip, cut a small hole toward bottom of each light bulb for stringing them together. Place on a parchment paper-lined baking sheet, leaving 2 inches between them. Repeat with second dough disk, rerolling scraps once. Freeze 15 minutes.

Bake in batches in preheated oven until light golden on edges and centers are set, 14 to 16 minutes. Cool on baking sheets 2 minutes. Transfer to wire racks to cool completely, about 20 minutes.

Prepare the Royal Icing: Combine all icing ingredients in the bowl of a heavy-duty electric stand mixer fitted with the whisk attachment. Beat on medium speed until thick and glossy, about 3 minutes.

Divide icing into small bowls, and stir in desired amounts of food coloring gel. (If not using icing immediately, cover with plastic wrap.) Transfer 1 color of icing to a piping bag, and pipe a small line around edges of Cookies to create borders. Thin remaining icing with a couple drops of water. Using a wooden pick, paintbrush, or offset spatula, cover Cookies with thinned icing, spreading all the way to piped border. Repeat with other colors and remaining light bulbs as desired.

Decorate with sprinkles, and brush bases of light bulbs with Luster Dust. Let stand until icing is dry, about 2 hours. Thread on decorative string or ribbon, if desired.