Preheat oven to 350 degrees F (190 degrees C).

Cream 1 cup of the butter with the 1 teaspoon nutmeg and the white sugar until light and fluffy. Beat in the egg, 1 tablespoon vanilla and 2 teaspoon rum flavoring. Gradually mix in the flour.

On a lightly floured surface, shape pieces of the dough into long rolls 1/2 inch in diameter. Cut the rolls in 3 inch lengths (to look like small logs). Place cookies on ungreased baking sheets.

Bake at 350 degrees F (175 degrees C) for 12 to 15 minutes or until light golden brown. Let cookies cool then spread with frosting on tops and sides. Mark frosting with the tines of a fork to resemble bark on a log. Sprinkle lightly with nutmeg.

To Make Frosting: Cream 3 tablespoons of the butter with 1 teaspoon of the vanilla and 1 teaspoon of the rum flavoring. Blend in 1/2 cup of the confectioners' sugar and beat well. Add the remainder of the confectioners' sugar alternately with the evaporated milk. Beat well after each addition and continue to beat until of spreading consistency.