Combine the butter and sugar in a large bowl and using an electric mixer at medium speed, beat until creamy and light. Add the egg yolks, vanilla and rum extracts, and continue beating until well mixed.
In a separate bowl, combine the flour, salt, nutmeg and cinnamon. Add to the butter and egg mixture and combine thoroughly to form a soft dough.
Working on a flat surface, divide dough in half and shape each half into a log about 8 inches long by 2 inches in diameter. Wrap tightly with plastic wrap and refrigerate for a minimum of 2 hours.
Preheat the oven to 350°F and line 2 baking sheets with parchment paper.
Remove the logs from the refrigerator and let them stand at room temperature for about 5 minutes. While still wrapped in plastic, roll the logs on a flat surface to round out the flat side that may have developed while they were chilling.
Unwrap, and using a sharp knife, cut the logs into 1/4-inch thick slices, turning the log a little after each cut to help keep it round.
Place the cookies 1 inch apart on the parchment lined sheets and bake for 10 to 14 minutes or until light golden in color. Cool on wire racks.
While the cookies bake, make the icing: Combine the sugar, butter and a pinch of salt in a large mixing bowl. Add the cream, rum, vanilla extract and nutmeg and beat with a mixer or by hand, until smooth and creamy.
After the cookies have cooled completely, coat the top of each with a thin layer of icing and sprinkle with a little extra nutmeg if desired. Allow the icing to harden completely before storing.
Store in an airtight container, no more than 2 layers deep, placing waxed paper between the layers.