To make the crust: Grease a 13 by 9-inch metal baking pan, then line with foil, leaving a 2-inch overhang on both ends, and grease foil.
Follow recipe for basic butter cookies to make dough (do not chill), then press dough evenly onto bottom of baking pan, using plastic wrap on top to prevent dough from sticking to your fingers, and chill until firm, about 20 minutes.
While crust chills, put oven rack in middle position and preheat oven to 375 degrees F.
Bake until golden brown, about 30 minutes. Cool crust in pan on a rack 20 minutes. (Leave oven on.)
To make brownie filling while crust cools: Melt butter and chocolate with brown sugar in a 3-quart heavy saucepan over moderate heat, stirring occasionally, until smooth. Remove from heat, then whisk in eggs and vanilla until combined well. Whisk in flour, cocoa, and salt, then stir in walnuts (if using).
Spread brownie batter over cooled crust and bake until set, about 20 minutes. Cool completely in pan on rack, about 1 1/2 hours.
Run a heavy knife under hot water and wipe dry, then cut confection into 1-inch squares. Brownie squares keep, layered between sheets of waxed paper or parchment, in an airtight container at room temperature 1 week.
Whisk together flour, baking powder, and salt in a small bowl.
Beat together butter and sugar in a large bowl with an electric mixer at medium-high speed until pale and fluffy, about 3 minutes in a standing mixer (preferably fitted with paddle attachment) or 6 with a handheld. Beat in egg and vanilla. Reduce speed to low, then add flour mixture and mix until just combined.