In medium bowl, whisk together flour, salt and nutmeg to combine; set aside. In large bowl, cream butter and sugar for 2 to 3 minutes or until light and fluffy. Add egg, rum extract and vanilla extract, mix until fully incorporated. Gradually stir in flour mixture until a soft dough forms.
Place 4 large sheets of plastic wrap onto work surface. Divide dough into 4 equal portions, placing each on sheet of plastic wrap. Roll each portion of dough into a log, 10-inches long by 1/2-inch thick, using plastic wrap to hold dough together and keep it from sticking to work surface. Wrap logs in plastic wrap; refrigerate at least two hours or until very firm. (Dough logs may be made up to 1 week ahead of time.)
Preheat oven to 350°F. Line three half sheet pans with parchment paper.
Working with one log at a time, unwrap dough and set on cutting board. Use a sharp paring knife to cut logs into 2-inch long pieces. Place cut pieces 2-inches apart on prepared baking sheets.
Bake 12 to 15 minutes or until lightly browned. Remove cookies and cool on baking sheet 2 minutes; transfer to wire rack to cool completely.
Meanwhile, prepare frosting. In medium bowl, cream together butter, rum flavoring, and vanilla extract. Gradually stir in one cup confectioner’s sugar. Continue stirring while slowly drizzling in one tablespoon heavy whipping cream. Blend in remaining one cup powdered sugar, followed by the last tablespoon of heavy whipping cream. Stir until frosting is smooth and creamy.
Use an offset knife or piping bag fitted with small star tip to frost cooled cookies. Garnish each with a sprinkling of freshly grated nutmeg.