In a medium bowl, whisk together the all-purpose flour and freshly grated nutmeg, setting it aside.

In a large mixing bowl, cream the softened butter with granulated sugar until light and fluffy, about 3-5 minutes. Add in the egg, vanilla extract, and rum extract, mixing well.

Gradually incorporate the dry flour mixture into the wet ingredients, forming a soft dough.

Take small portions of the dough and roll them into logs about ½ inch in diameter. Cut each log into approximately 2-inch pieces and place them on an ungreased baking sheet.

Preheat your oven to 350°F (175°C). Bake the Nutmeg Logs for 12-15 minutes or until lightly golden at the edges.

While the cookies cool, prepare the icing by whisking together powdered sugar, milk, vanilla extract, and rum extract in a medium bowl until smooth.

Allow the Nutmeg Logs to cool completely on a wire rack before icing. Drizzle icing generously over each cookie.

Sprinkle a dash of nutmeg or festive sprinkles on top before the icing sets.