Preheat oven to 325 degrees. Butter a 9-inch round tart pan with a removable bottom; set aside. Pulse walnuts in a food processor until finely ground. Transfer to a bowl, and whisk in flour and salt.
In the bowl of an electric mixer fitted with the paddle attachment, beat butter, sugar, and vanilla until pale and fluffy. Slowly add walnut mixture, and beat until just combined.
Transfer dough to prepared pan. Place a piece of plastic wrap on dough, and use it to press the dough evenly into pan (this will prevent the dough from sticking to your fingers). Transfer to refrigerator and chill until firm.
Cut dough into 8 wedges using a paring knife. Prick all over with a wooden skewer or fork.
Bake until golden brown and firm in the center, 40 to 45 minutes. Transfer pan to a wire rack. Recut into wedges and prick again. Let cool.