Delicate and thin, these coconut sugar cookies are cut into wintry shapes and whimsically decorated.
Process coconut in a food processor until finely ground. Whisk together flour, coconut, baking powder, and salt in a small bowl; set aside. Put butter and granulated sugar into the bowl of an electric mixer fitted with the paddle attachment; mix on medium speed until pale and fluffy, about 3 minutes. Reduce speed to low. Mix in eggs, vanilla, and coconut extract, if desired. Add flour mixture; mix until just combined. Divide dough into thirds; wrap each in plastic. Refrigerate until firm but malleable, about 1 1/2 hours.
Roll out dough between 2 sheets of lightly floured parchment to 1/4 inch thick. Repeat, and stack between parchment on a baking sheet. Freeze or refrigerate dough until firm, 30 minutes.
Preheat oven to 350 degrees. Cut out shapes with a floured cutter. Space 11/2 inches apart on baking sheets lined with parchment. To decorate with sugar, lightly brush with water and sprinkle with sugar.
Bake until edges are golden, 11 to 13 minutes. Let cool on sheets 3 minutes. Transfer to wire racks; let cool completely. If frosting cookies, put glaze in a small pastry bag fitted with a small plain round tip (such as Ateco #6), and pipe onto cookies. Alternatively, spread glaze on cookies using a small offset spatula. Let glaze dry completely, or sprinkle with sugar or sprinkles when still wet then let dry completely, at least 2 hours.
Unfrosted cookies can be stored in airtight containers at room temperature up to 1 week.