In a medium bowl, combine the all-purpose flour and freshly grated nutmeg, then set aside.

In a separate large mixing bowl, cream together softened butter, granulated sugar, vanilla extract, and rum extract until light and fluffy, about 3-4 minutes.

Add the egg and mix until fully incorporated. Gradually fold in the dry ingredients until a soft dough forms.

Turn the dough onto a lightly floured surface, divide it into portions, and roll each into logs about ½-inch in diameter. Cut into 2-inch pieces and shape uniformly.

Place the cookie pieces on an ungreased baking sheet, leaving space for spreading. Preheat oven to 350°F (175°C).

Bake for 12-15 minutes until edges are light golden and centers remain soft. Let cool on the baking sheet for a few minutes before transferring to a wire rack.

Prepare the icing by whisking together powdered sugar, milk, and a splash of vanilla and rum extract until smooth.

Spread the icing over the cooled cookies and lightly dust with nutmeg while the icing is still wet. Allow to set before serving.