Sift together flour, baking powder, and salt.
Cream butter and add both sugars gradually, creaming thoroughly until light and fluffy.
Add the egg and vanilla and mix until fully incorporated.
Stir in the walnuts.
Add flour gradually, mixing after each addition, until just incorporated. .
Split the dough half. Place each portion on a sheet of parchment paper, waxed paper, or plastic wrap and shape into rolls about 1½ inches in diameter.
Roll up and refrigerate for several hours until firm. You can also pop them in the freezer to speed up the process.
When ready to bake, preheat the oven to 350°F.
Line sheet pans with parchment paper and set aside.
Slice into ½-inch rounds and place on the prepared baking sheets.
Bake for 10–12 minutes, until the cookies are just firm and lightly golden.
Rest in the pan for 5 minutes before transferring to a wire rack to cool.