COMBINE flour, 2/3 cup icing sugar and salt in a food processor. Pulse until just mixed. Add butter. Pulse until a soft dough forms. Gather dough into a ball, then divide in half. Shape each half into a flat disc and wrap separately in plastic wrap. Refrigerate until firm, about 1 1/2 hours, or over-wrap with heavy foil and freeze up to 1 month.
POSITION oven racks in top and bottom thirds of oven. Preheat oven to 325F. (Keep discs in refrigerator until ready to roll.) Dust a rolling pin and a large sheet of waxed paper lightly with flour. Roll out a disc of dough on waxed paper to no thicker than 1/8 in. Dip a 2-in. rectangular cookie cutter in flour. Cut out shapes. Transfer to 2 large, ungreased cookie sheets and place about 1 in. apart. Bake in top and bottom thirds of oven, switching and rotating sheets halfway through, until cookies are just golden around edges, 10 to 12 min.
REMOVE to a rack to cool completely. Repeat with remaining dough. Gather dough scraps and form into a disc. Chill until firm enough to re-roll, about 10 min.
WHISK maple syrup with 1/2 cup icing sugar until smooth. Lightly brush tops of cooled cookies, covering the entire surface. Excess icing will drip through the racks. Let stand for 1 hour or until icing is dry. Store cookies in an airtight container at room temperature up to 1 week or freeze up to 1 month.
Prep Tip: If shortbread cookie dough softens while you’re rolling and cutting it out, simply stop and refrigerate it on waxed paper, and check on it after 5 min.