Line a large baking sheet with parchment paper. Place the butter, sugar and vanilla in a medium bowl and beat until well combined. Mix in the egg white, and then beat in the flour and walnuts until well combined.
Roll the dough into 1" balls and set on the baking sheet, spacing each cookie about 2" apart. Press each cookie into 1/2" thick disc. Chill in the fridge for 10 minutes.
Place an oven rack in the middle position. Preheat the oven to 350 degrees F. Bake the cookies 15 minutes, or until pale golden around the edges. Cool on a baking rack for 5 minutes, and then, while still warm, coat in the icing sugar, shaking off the excess.
Note: A food processor can be used to finely crush the walnuts. If you don’t have one, place the walnuts in thick a plastic bag and use a kitchen hammer to crush them.