Heat oven to 350 degrees. Prepare cake recipe in an 11-by-17-inch baking pan (prepare pan as recipe directs). Bake until cake springs back when pressed, about 40 minutes. Remove from oven; let cool.
Line two 11-by-17-inch baking sheets with parchment. Using an offset spatula, spread the ice cream inside a 3-inch-diameter round cookie cutter. Place the ice-cream disk on a baking sheet. You will need 12 ice-cream disks. You may have to work in batches, placing the ice cream back in the freezer if it becomes too soft. Store completed ice-cream disks in the freezer until ready to assemble.
Using the same 3-inch round cookie cutter, cut out 12 rounds of the yellow cake. Place an ice-cream round on each larger cake round, and return the assembled cakes to the baking sheets. Return the baking sheets to the freezer until ready to assemble cakes.
Using a 1 5/8-inch-diameter round cookie cutter, cut 18 more circles from the cake; cut 6 circles in half crosswise, and place one half on each full-size round; set aside, tightly wrapped in plastic.
Carefully crack eggs in half, reserving yolks and whites for another use. Wash and dry shells thoroughly. Set aside.
Transfer the prepared meringue to a pastry bag fitted with desired tip. Remove the ice-cream-topped cakes from the freezer. Top each with one of the reserved smaller cake stacks, and secure by placing a dab of meringue between the ice cream and the cake that covers it. Quickly pipe meringue over each cake entirely. Press eggshell cups into tops of cakes. Immediately place into freezer for at least 3 hours, or overnight.
Heat oven to 500 degrees. Bake cakes until lightly browned, about 2 minutes. Remove from oven, and transfer at once to chilled serving plates. Working quickly, fill each eggshell cup with 1 teaspoon liquor, and ignite. Serve immediately.
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