Whisk together flour and nutmeg in a medium bowl. Cream together butter and sugar in a stand mixer until light and fluffy, about 2-3 minutes. Add vanilla and rum extracts along with the egg, mixing until combined. Gradually add the dry mixture, stirring until a smooth dough forms.

Turn the dough onto a lightly floured surface, divide into four portions. Roll each into a log about 1/2 inch in diameter. Cut each log into 2-inch pieces and place on an ungreased baking sheet.

Preheat oven to 350°F (175°C). Bake cookies for 12-15 minutes until lightly golden around the edges. Allow cooling on the baking sheet for a few minutes before transferring to a wire rack.

Prepare the icing by whisking together powdered sugar, milk, vanilla, and rum extracts until smooth. Adjust thickness if necessary.

Frost cooled cookies with icing and add a sprinkle of nutmeg or festive sprinkles. Allow icing to set before serving.