In a food processor or bowl, combine flour, 1/4 cup sugar, and salt. Add butter and whirl or cut in with a pastry cutter until fine crumbs form. In a small bowl, beat egg with vanilla to blend. Add to flour mixture and whirl or mix with a fork until dough is evenly moistened and starts to clump together. Press into a ball, then divide in half.
On a 12- by 15-inch piece of plastic wrap, using your hands, roll one dough half into a 6-inch-long log; cover with another piece of plastic wrap. With a rolling pin, roll log into an even, 1/4-inch-thick rectangle about 6 inches wide and 9 inches long (edges can be ragged). Repeat to roll remaining dough half. Transfer each rectangle to a 14- by 17-inch baking sheet and chill until firm, 15 to 20 minutes.
Working with one rectangle at a time (keep other chilled), transfer to a board. Pull off top sheet of plastic wrap and cut dough crosswise into 16 strips about 1/2 inch wide and 6 inches long. On a 14- by 17-inch baking sheet lined with cooking parchment, or a buttered nonstick baking sheet, arrange four strips in a tic-tac-toe pattern: two parallel, then two over the top, perpendicular to the first, about 1 inch apart. Press lightly where strips cross. Repeat to shape remaining cookies, spacing at least 1/2 inch apart on baking sheets (you'll need two). Sprinkle with 2 to 3 tablespoons sugar.
Bake cookies in a 325° regular or convection oven until lightly browned at edges, 20 to 25 minutes; switch sheet positions halfway through baking. Transfer sheets to racks to cool. With a wide spatula, gently loosen cookies.
Nutritional analysis per cookie.