In a large bowl, cream butter and granulated sugar until light and fluffy. Beat in vanilla.
In a separate bowl, combine flours, corn starch, and coconut. Add to butter mixture and mix until blended. Shape dough into a flattened disc. Wrap in plastic and refrigerate dough for 30 minutes if dough is too soft to roll.
On a floured surface, roll out dough until 1/4 inch thick, dusting lightly with flour as necessary. Cut dough into rounds with a 2 inch cookie cutter dipped in flour. Reroll dough as necessary, without adding too much flour. Place cookies on parchment-lined baking sheets. Sprinkle lightly with sugar.
Bake cookies in a preheated 300F oven for 25 minutes, or until just beginning to turn golden. Cool on wire racks.