Position racks in the upper and lower thirds of the oven and preheat to 325 degrees . Line 2 cookie sheets with parchment paper. Using a food processor, grind the walnuts with 1 tablespoon sugar. Transfer to a bowl. Add the remaining sugar to the processor and mix until superfine, 1 minute. Add the flour, salt and ground walnuts; pulse to combine. Mix in the butter and vanilla until crumbly. Transfer the dough to a work surface, shape into a ball, halve it and form into 2 disks. Wrap with plastic wrap and refrigerate for 10 minutes.
Place a dough disk on each cookie sheet and, using your hands, shape each into an 8-inch round. Cut each round into 12 wedges and leave intact. Bake, switching the pans halfway through, until the edges are browned, about 35 minutes. Recut the hot shortbread into wedges, separate lightly and let cool. (Alternatively, shape each disk into a rectangle and cut each rectangle into bars, then proceed with baking.)