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For the cookies:
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1. Combine flour, salt, and nutmeg in a small bowl and whisk to combine. Set aside.
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2. In a large mixing bowl, cream together butter and sugar until fluffy.
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3. Add the egg, rum flavoring, and vanilla extract, stirring until well incorporated.
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4. Add the flour mixture into the creamed mixture in 3 separate additions, stirring after each addition to combine.
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5. Divide dough into 4 equal portions, and roll each portion into a 10-inch long log, about 1/2-inch thick. Wrap logs in plastic wrap and refrigerate for several hours, or up to 1 week.
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6. When you are ready to make the cookies... Preheat the oven to 350 degrees F. Line baking sheets with parchment or a silicone baking mat.
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7. Cut logs into 2-inch pieces. Place pieces on the prepared baking sheets, about 2 inches apart.
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8. Bake for 12 to 15 minutes, or until lightly browned.
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9. Remove from oven, and allow cookies to cool for 2 minutes on the baking sheet before transferring to a wire rack to cool completely.
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While cookies are cooling, make the frosting:
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1. Cream together the unsalted butter, rum flavoring, vanilla extract, and milk. Add the powdered sugar, and stir until blended and creamy. Frost the cooled cookies using a knife or you can pipe the frosting onto the cookies using a pastry bag.
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2. Sprinkle each cookie with a touch of nutmeg, for garnish.
