Preheat the oven to 325°F and line the pans with parchment paper leaving an overhang on two sides of each pan.
Combine the butter, sugar, cocoa powder and salt in a heatproof bowl or saucepot. Set the bowl atop a saucepot of simmering water to create a double-boiler. Stir the butter from time to time until it's melted and the mixture is smooth and hot to the touch. Remove the bowl from the double-boiler and set it aside to cool for 10 minutes.
Using a wooden spoon or rubber spatula, stir in the vanilla and then stir in the eggs, one at a time, stirring vigorously between each addition. Once the batter looks well-blended, shiny and thick, stir in the flour until it's fully incorporated. Spread half of the batter into each of the prepared baking pans.
Bake the brownies for 10 to 12 minutes, until a toothpick inserted comes out with a few crumbs. (The brownies should be thin and slightly underbaked so that when they freeze they aren't too difficult to bite into.) Remove the brownies from the oven and let them cool completely (still in the pans) on a wrack.
Microwave the ice cream in 10-second intervals, stirring between each interval, until it's slighty softened, about 20 seconds total. (See Kelly's Notes.)
With the brownies still in the pan, spread the softened ice cream on top of one pan of brownies. Dollop the peanut butter on top of the ice cream, and then using a knife or spatula, swirl the ice cream and peanut butter together. Using the overhangs on the second pan, life the brownies out of the pan and transfer them on top of the ice cream top to form a large sandwich. Cover the pan in plastic wrap and freeze it for at least 3 hours, until the ice cream is firm.
Once firm, unwrap and remove the brownie to a cutting board. Using a serrated knife, slice the brownie in half widthwise and then in fourths lengthwise to form 8 ice cream sandwiches. Serve the ice cream sandwiches immediately or wrap them individually and freeze them until ready to serve.