In a small saucepan, combine 1/2 cup sugar and 2 1/2 tablespoons water and bring to a boil. Cook until sugar is dissolved, then remove from heat and let cool. When syrup has cooled to room temperature, add half the liqueur (or as needed) to bring volume up to 1/2 cup. Transfer to an airtight container and refrigerate or place in freezer until thoroughly chilled.

Lightly butter a 9-by-5-by-3-inch loaf pan and line with parchment paper, leaving a 2- to 3-inch overhanging border on all sides. Using a rubber spatula or the back of a spoon, spread half of the first ice cream into bottom of pan, spreading to fill corners evenly. (The ice cream should fill about 1/3 of the pan.) Cut a piece of brownie the same size as the pan and press firmly onto top of ice cream. Using a spoon or pastry brush, drizzle half the chilled syrup over brownie. Top with sorbet or contrasting ice cream flavor, smoothing to fill corners of pan, then follow with remaining first flavor ice cream, again smoothing to fill pan corners. Top with another layer of brownie cut to the same size as pan, and drizzle with remaining syrup. Fold overhanging parchment over top of cake and wrap entire cake pan with plastic wrap. Transfer to freezer for at least 4 hours and up to several days; it is important that this is frozen solid before proceeding.

Place egg whites in the bowl of a standing mixer and beat until foamy with slightly soft peaks. Turn off mixer. Combine remaining 1 cup sugar with 1/3 cup water in a small heavy saucepan and cook over medium-high heat until syrup registers 250 degrees on a candy thermometer. Begin beating egg whites again on medium-high. Slowly drizzle hot syrup carefully down side of bowl, being careful not to pour hot syrup on beaters or it will spatter. Continue beating egg whites until a stiff and a glossy meringue is formed and outside of mixing bowl is barely warm, about 5 minutes.

Remove frozen layered ice-cream cake from freezer and place on a rectangular serving platter lined with aluminum foil. Place meringue in a pastry bag fitted with a 3/4-inch star tip and decoratively cover entire cake with meringue. Transfer to freezer for 15 minutes.

Remove cake from freezer and make an indentation with a tablespoon in center of meringue deep enough to fit eggshell. Using a kitchen torch, brown meringue lightly all over. (Alternately, instead of browning with a torch, preheat oven to 450 degrees. Freeze dessert for 15 minutes after decorating with meringue, then bake for 2 to 3 minutes, just until meringue takes on a bit of color.)

Place eggshell in indentation. Pour remaining 2 tablespoons of rum into eggshell and carefully ignite with a kitchen match. Spoon flaming rum evenly over meringue (flaming rum will brown meringue a little more). Remove eggshell and serve baked Alaska immediately.

Cook's Note: Propane gas torches are highly flammable and should be kept away from heat, open flame, and prolonged exposure to sunlight. They should be used only in well-ventilated areas. When lighting a propane gas torch, place the torch on a flat, steady surface, facing away from you. Light a match or lighter and then open the gas valve. Light the gas jet and blow out the match. Always turn off the burner valve to "finger tight" when finished using the torch. Children should never use a propane gas torch without adult supervision.