Whisk together flour, baking powder, salt, cinnamon and nutmeg; set aside. In separate large bowl, beat together butter and sugar until light and fluffy. Beat in egg, coconut milk and vanilla. Stir in flour mixture; beat just until blended. Shape dough into disk; wrap in plastic wrap and refrigerate for at least 2 hours or overnight.
Preheat oven to 350˚F. On lightly floured surface, roll dough into 1/8-inch thickness. Using floured 3-inch round cookie cutters, cut out rounds. Place cookies on parchment paper–lined cookie sheets, about 2 inches apart (reroll dough as needed, chilling dough if too soft to roll).
Bake cookies, in batches, for 8 to 10 minutes or until tops are set and bottoms are lightly golden. Let stand for a few minutes in pan on rack. Let cool completely.
In bowl, whisk together icing sugar, coconut milk, cream of coconut, rum, cinnamon and salt.
Dip top of cookies in icing, letting excess drip back into bowl. Let stand for 15 to 20 minutes or until icing is set. Store in airtight container for up to 1 week or freeze for up to 1 month.
Sprinkle iced cookies with toasted coconut if desired.
Substitute brandy or whisky for rum. Alternatively, for an alcohol-free glaze, substitute 1 tsp. vanilla extract.