Adjust oven racks to the middle of the oven. Preheat the oven to 350 degrees.

Beat the softened butter until light. Gradually add 1 cup sugar and the egg and beat several minutes, until pale and very fluffy. Beat in the vanilla. Gradually stir in the flour and nuts, blending well to make a stiff dough. Gather the dough into a ball. There is no need to refrigerate.

Divide the dough in quarters and roll each quarter separately to a thickness slightly more than 1/4 inch on a lightly floured surface with a lightly floured rolling pin. Cut into rounds with a 1-1/2-inch cookie cutter. Place the rounds on ungreased cookie sheets, 1/4 inch apart. Reroll and cut out all the trimmings.

Bake for 15 minutes, or until the bottoms of the cookies are lightly browned and the edges just begin to brown.

Dip the warm cookies on both sides in additional sugar. Place the cookies on wire racks to cool thoroughly.

Store in tightly covered containers for up to a week at room temperature, or freeze. If frozen, let them defrost and return to room temperature before serving.