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Preheat the oven to 350 degrees.
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Whisk the flour, nutmeg, and salt together in a medium bowl. (These are the dry ingredients.)
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In a large bowl, beat the butter with an electric mixer until very creamy. Add the sugar and beat until light and fluffy. Beat in the vanilla, rum extract, and egg.
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With mixer on low, gradually beat in the dry ingredients. Mix on medium speed until a cohesive dough forms; the dough will be stiff.
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Roll balls of dough into ropes about a 1/2-inch in diameter, then slice each rope into 2 1/2-inch lengths. (If necessary, you can dust a little flour on your counters to help you shape the dough; I find the dough is stiff enough to roll without a floured surface.)
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Arrange 1 inch apart on un-greased, unlined baking sheets. Bake 12 to 18 minutes, until lightly browned on the bottom.
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Cool the logs completely on wire racks before frosting them.