In a large bowl, whisk together butter and sugar until creamy. Add eggs, one at a time, and beat until smooth. Beat in vanilla. Add the flour mixture and beat until combined. (The batter will be thick.)
Scrape batter into the pan and, using an offset spatula, smooth to a thin, even layer, leaving a small border of parchment paper around the edges. Bake until cake is set, 10 to 12 minutes. Transfer pan to a rack to cool completely.
When you are ready to assemble, remove ice cream from freezer and let it stand for a few minutes to soften slightly.
Using the parchment, transfer the cake to a cutting board. Cut cake in half crosswise so that you have 2 pieces that are about 9 by 13 inches each. Using a large ice cream scoop, dollop ice cream evenly over the top of one half of the cake. Using an offset spatula, spread ice cream into an even layer. Place the remaining piece of cake on the ice cream, with the top facing down, to make a large sandwich cake. Immediately wrap it well in plastic wrap and freeze until firm, at least 8 hours.
Remove the large sandwich cake from the freezer and take off the plastic wrap. Trim about 1/4 inch off of each edge and then cut into 12 smaller sandwiches. Wrap each sandwich well in plastic wrap and freeze until firm and ready to serve.
Mothers Secret Recipes of Comfort Food & Culinary Inspiration from an Organic and Allergen-free Approach