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 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 1 1/2 inches apart on baking sheets lined with parchment. To decorate with sugar, lightly brush with water and sprinkle with sugar. Or once cooled, decorate with icing and sprinkles as desired.
Bake until edges are golden, 11 to 13 minutes. Let cool on sheets 3 minutes. Transfer to wire racks; let cool completely.