Nonstick vegetable oil spray
1¼ cup (2½ sticks) unsalted butter, divided
¾ cup (50 g) unsweetened cocoa powder
½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt
½ tsp. baking powder
2 cups granulated sugar, divided
1 tsp. vanilla extract
3 large eggs
1¾ cups (219 g) all-purpose flour, divided
¼ cup creamy peanut butter or other nut butter
Flaky sea salt
3 pints vanilla ice cream, softened on counter about 5 minutes
Place a rack in center of oven and preheat to 350°F. Generously coat a 18x13" rimmed baking sheet with nonstick spray . Melt 1 cup (2 sticks) butter in a small saucepan over medium heat. Let cool slightly.
Whisk together cocoa powder, kosher salt, baking powder, and 1½ cups sugar in a medium bowl. Drizzle in butter; reserve saucepan. Whisk until butter is incorporated, then add vanilla and 2 eggs and whisk until batter is glossy and smooth. Add 1¼ cup flour and whisk until no dry streaks remain.
Melt remaining ¼ cup (½ stick) butter in reserved saucepan over medium heat. Remove from heat, add peanut butter, and whisk until smooth. Whisk in remaining 1 egg and ½ cup sugar, then add remaining ½ cup flour and whisk until smooth.
Pour chocolate batter into prepared pan and, using a rubber spatula or your fingers, smooth into an even layer, spreading all the way to the corners and sides; the batter will be quite thin. Dollop peanut butter batter over chocolate batter and press into an even layer. Sprinkle sea salt on top.
Bake cookies until top looks dry and a tester inserted into the center comes out clean, 14–17 minutes (cookies will seem too soft, but they’ll firm up as they cool). Let cookies cool in baking sheet, then use a sharp knife to cut into 32 pieces (8 across on long sides and 4 down on short sides). Carefully remove pieces from pan with a thin spatula.
Meanwhile, line a rimmed 13x9" rimmed baking sheet or baking dish with plastic wrap, layering sheets both lengthwise and crosswise and leaving a few inches of overhang on all sides. Transfer softened ice cream to prepared pan and spread into an even layer. Fold overhang up and over the top and freeze ice cream until firm, at least 1 hour.
When you’re ready to assemble sandwiches, use plastic overhang to transfer frozen ice cream block to a cutting board. Working quickly, cut into 16 pieces (4 across on long sides and 4 down on short sides—mirroring half of the cookie slicing method above). Sandwich each ice cream rectangle between 2 cookies, arranging cookies top side out and place on a baking sheet. Freeze until firm all the way through, then transfer sandwiches to an airtight container or individually wrap in foil. Keep frozen until ready to serve.