Preheat the oven to 350°F (175°C) and line two baking sheets with parchment paper.
In a medium bowl, mix the flour, baking soda, baking powder, nutmeg, cinnamon, and salt. In another bowl, cream the butter, granulated sugar, and brown sugar until fluffy, then add egg, vanilla, and eggnog.
Gradually mix the dry ingredients into the wet ingredients until combined.
Drop rounded spoonfuls of dough onto prepared baking sheets.
Bake for 10-12 minutes until lightly golden. Cool on the sheet for 5 minutes before transferring to a wire rack.
Prepare the glaze by whisking together powdered sugar and eggnog. Drizzle over cooled cookies and let set before serving.
Notes
Store in an airtight container at room temperature for up to a week, or freeze for longer storage.