In a medium bowl, whisk together the all-purpose flour and freshly grated nutmeg until well combined. In a separate bowl, cream together the softened unsalted butter, granulated sugar, vanilla extract, and rum extract using an electric mixer on medium speed for about 3-4 minutes, or until light and fluffy. Crack in the egg and mix until fully incorporated, then gradually add the dry ingredients until a smooth dough forms.

Lightly flour a clean surface and take small portions of the dough to roll into ½-inch thick logs. Cut the logs into 2-inch pieces and place them about 2 inches apart on an ungreased baking sheet.

Preheat your oven to 350°F (175°C). Bake the cookies for 12-15 minutes, watching closely as they approach the 12-minute mark. They should be lightly golden at the edges while staying soft in the center.

Once baked, cool the nutmeg logs on the baking sheet for about 5 minutes before transferring them to a wire rack. Prepare the icing by whisking together powdered sugar, milk, vanilla extract, and optional rum extract until smooth.

After cooling, frost each cookie generously with icing and sprinkle with extra grated nutmeg or holiday sprinkles before letting the icing set.