In a bowl, with an electric mixer on medium speed, beat butter and sugar until smooth. Stir or beat in flour and coconut until well blended.
On cooking parchment-lined 12- by 15-inch baking sheets (see notes), shape 1/2-cup portions of dough into 3/4-inch-thick logs, each about 13 inches long; place about 3 inches apart. Press your fingers across each log to make ridges, flattening logs to about 1/8 inch in thinnest parts.
Bake logs in a 400° oven until edges are lightly browned, 8 to 10 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. Slide a thin spatula under logs to loosen.
In a small bowl, mix powdered sugar and rum until smooth. Brush warm logs with powdered sugar mixture, then immediately cut diagonally into 1-inch-wide strips. Cool cookies completely on pans.
Note: Nutritional analysis is per cookie.