Line the bottom and sides of a 9-by-13-inch baking pan with aluminum foil and lightly grease the foil with the tablespoon of softened butter.
Put the rest of the butter in a large saucepan over medium heat. Stir with a whisk until the butter melts.
Using a pot holder, remove the saucepan from the heat and add in the cocoa powder. Whisk the butter and cocoa powder until the mixture is smooth and no lumps remain.
Add the sugar while continuing to whisk the mixture until it is well blended. Let the mixture cool for 2 minutes.
Add the eggs and vanilla. Whisk until well blended.
Add the flour and salt and stir until the batter is blended. Scrape down the sides of the bowl with a rubber spatula as needed.
Pour the batter onto the prepared baking pan and spread evenly.
Bake the batter until a toothpick inserted into the center comes out clean. This takes about 20 minutes.
Remove the pan from the oven, using oven mitts and set the pan on a wire rack. Let cool completely.
Using the ends of the foil, lift the baked cake from the pan and set it on a cutting board.
Cut the cake in half lengthwise and carefully loosen both halves from the foil.
Using a rubber spatula, spread the softened ice cream evenly over one half of the cake into a layer about 1 inch thick.
Top with the other half of the cake and press down gently. You may find the easier way to do this is to cut the foil in half and place the thin cake layer on top of the ice cream without attempting to remove the cake and handle it. This can precent the thin layer of cake from breaking apart.
Wrap the ice cream sandwiches in the foil and freeze until hard, about 6 hours.
Peel away the foil and cut into sandwiches.
To store the extra ice cream sandwiches, wrap them in plastic wrap and place them back in the freezer. They will keep for up to 2 weeks.