In a bowl, with an electric mixer on medium speed, beat butter, granulated sugar, and brown sugar until smooth. Beat in egg and orange peel until well blended.
In another bowl, mix flour, walnuts, baking soda, and salt. Stir or beat into butter mixture until well blended.
Divide dough in half; form each portion into a 2 1/2-inch-thick log. Wrap logs airtight in waxed paper. Freeze until firm enough to slice neatly, at least 2 hours, or up to 3 weeks.
Unwrap dough. Using a sharp knife, cut logs crosswise into 1/8-inch-thick slices. Lay slices 3 inches apart on buttered or cooking parchment–lined 12- by 15-inch baking sheets; if cookies flattened when cut, reshape with your fingers.
Bake cookies in a 375° oven until edges are very lightly browned, about 12 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. With a wide spatula, transfer cookies to racks to cool completely.
Nutritional analysis per cookie.