With an electric mixer, beat butter, sugar, and salt until smooth. Mix in flour, then coconut, beating until a dough forms. Transfer to a piece of waxed paper; pat into a rectangular log, about 3 inches wide and 6 inches long. Wrap with waxed paper; freeze until firm, about 30 minutes.
Preheat oven to 350 degrees, with racks in upper and lower thirds. With a serrated knife, slice log of dough crosswise 1/4 inch thick (you should have 24 slices); arrange slices on two baking sheets.
Bake until golden, rotating sheets from top to bottom and front to back halfway through, 20 to 25 minutes (watch closely toward end of cooking time to avoid overbrowning). Cool completely on sheets.
Dividing evenly, spread ice cream on flat side of half the cookies; sandwich with remaining cookies, flat side down. Freeze on a baking sheet until firm, about 3 hours.
Slicing cookies from a log of dough (instead of rolling out and cutting) is an easy way to ensure they are of equal size for perfect sandwiches.