Preheat the oven to 350 degrees F and position the oven racks in the middle of the oven. Lightly grease 2 large baking sheets with 1 tablespoon of the butter and set aside.
Sift the flour, baking powder, baking soda, and salt into a medium bowl. Place the butter and sugar in the bowl of an electric mixer and cream the ingredients on high speed. Add the vanilla and egg, and combine on medium speed. Stop the mixer and scrape down the sides of the bowl. Add the flour mixture and combine on low speed. Remove bowl from mixer. Stir in walnuts by hand.
Form into balls about 2 tablespoons each in size. Divide the balls of dough between the prepared baking sheets, and press to slightly flatten, leaving about 2-inches of space between each cookie. Bake in batches in the oven until golden brown, about 20 minutes. Remove from the oven and cool on wire racks.
Yield: 24 cookies