In a mixing bowl, whisk together the all-purpose flour and freshly grated nutmeg. In a separate bowl, cream the butter and granulated sugar until light and fluffy. Add vanilla and rum extracts, then the egg, mixing until fully incorporated. Gradually combine the wet and dry ingredients until a soft dough forms.

Lightly flour your work surface and turn the prepared dough onto it. Divide the dough into sections, rolling each into logs approximately 1/2 inch in diameter. Cut into 2-inch pieces and arrange on an ungreased baking sheet.

Preheat your oven to 350°F (175°C). Bake the cookies for 12-15 minutes, removing when just golden around the edges to maintain softness.

Prepare the icing by whisking powdered sugar, milk, vanilla extract, and rum extract until smooth. Adjust the consistency to be drizzleable.

Once cooled, frost each cookie with icing and sprinkle with nutmeg or holiday-themed sprinkles.