In a large bowl, cream butter and sugar until light and fluffy. Beat in egg, 1/3 cup eggnog and 1 teaspoon extract. In another bowl, whisk flour, baking powder, cinnamon, nutmeg, salt, ginger and allspice; gradually beat into creamed mixture.
Divide dough in half and shape each portion into a disk; cover and refrigerate until firm enough to roll, about 30 minutes.
Preheat oven to 375°. On a lightly floured surface, roll each portion of dough to 1/4-in. thickness. Cut with a floured 3-1/4-in. star-shaped cookie cutter. Place 1 in. apart on parchment paper-lined baking sheets.
Bake until edges begin to brown, 8-10 minutes. Cool on pans 1 minute. Remove to wire racks to cool completely. For glaze, mix confectioners’ sugar, remaining extract and enough remaining eggnog to achieve a drizzling consistency; drizzle over cookies. Decorate as desired.
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