In a medium mixing bowl, whisk together 2 cups of all-purpose flour and 1 teaspoon of freshly grated nutmeg. In a separate bowl, cream together 1 cup of unsalted butter and ¾ cup of granulated sugar until light and fluffy, approximately 3-4 minutes. Stir in 1 teaspoon of vanilla extract and 1 teaspoon of rum extract. Add in the egg and mix until well combined, then gradually incorporate the dry ingredients to form a soft dough.

Transfer the dough onto a floured surface and divide it into manageable portions. Roll each portion into logs about ½ inch in diameter, ensuring they are even throughout. Cut the logs into 2-inch pieces to create uniform cookies. Place the shaped cookies on ungreased baking sheets, leaving space between them for even baking.

Preheat your oven to 350°F (175°C). Once preheated, bake the Nutmeg Logs for 12-15 minutes, until they are just set and lightly golden around the edges.

While the Nutmeg Logs cool, prepare the icing by whisking together 1 cup of powdered sugar and 2 tablespoons of milk in a small bowl until smooth. Adjust the consistency if needed.

Ensure that the Nutmeg Logs are completely cooled before frosting. Use a fork or piping bag to drizzle the icing over each cookie decoratively. Allow the icing to set before serving or storing.