To make walnut brownies: Butter a 12-by-15-inch jelly-roll pan. Line bottom of pan with a parchment paper rectangle cut to fit; butter parchment. Lightly dust pan with flour.
Preheat oven to 350 degrees F. Bring about 2 inches water to a boil in a saucepan or the bottom of a double boiler; remove from heat. Combine butter and chocolate in a heat-proof bowl or in the top of double boiler. Set bowl over hot water and let stand, stirring occasionally with a flexible spatula, until melted and smooth, about 15 minutes. In another bowl, whisk eggs, sugar, and vanilla just to blend. Stir melted chocolate into egg mixture until incorporated. In another bowl, stir together flour, baking powder, and salt. Stir flour, baking powder, and salt into chocolate mixture until blended.
Spread batter evenly in prepared pan and sprinkle walnuts over the top. Bake until edges feel firm and a skewer inserted into the center comes out with moist crumbs attached, 18 to 20 minutes. Cool brownie completely in pan.
Run a knife along the inside edge of pan to release brownie. Place a large piece of aluminum foil on a flat work surface. Invert pan over foil to release brownie. Trim about 1/4 inch from the edges, and then cut brownie in half lengthwise down the center to make 2 rectangles, each about 10 by 7 inches.
Soften ice cream. Spread the ice cream gently and evenly over one of the brownie halves. Place the other brownie half, walnut-side up, on top of the ice cream; gently compress to flatten the sandwich and bring the ice cream out to the edges. Wrap the foil securely around the brownie and freeze until firm, at least 8 hours or up to 2 days.
Remove brownie from freezer and remove foil. Using a sharp knife, cut into 12 rectangles. Serve sandwiches immediately, or wrap individually in plastic wrap and freeze up to 2 days.
Recipe from Ice Cream Treats by Charity Ferriera/Chronicle Books, 2011.