Arrange the oven rack to the center position in the oven. Preheat oven to 350°. Line a baking sheet with parchment paper or a silpat. Brush the parchment/silpat with a little butter and set aside.
Melt the butter over medium heat in a saucepan. Stir in the sugar and remove from heat. Add rum, egg, salt and beat with a wooden spoon until smooth. Stir in the flour, chopped nuts and flaked coconut.
Spread the batter in 1 teaspoon rounds onto the prepared baking sheet, leaving 2-3 inches space between the cookies as they will spread. I recommend baking no more than 4 cookies at a time as they can be a little persnickety when removing them from the pan.
Bake for 4 minutes, then turn the pan 180° in the oven and cook for an additional 3-5 minutes or until cookies are golden. Remove from oven and let rest for 30 seconds before carefully sliding the offset spatula under the cookies to remove them from the pan. Transfer the cookies to the rolling pin to form the curve. Allow the cookies to cool on the rolling pin until they have crisped.
Brush the pan with a little more butter and continue to bake the cookies in the same fashion described above.