There's a reason baked Alaska always seems rank high on the list of desserts people dread making: It's kind of a pain. The first step alone can be daunting: Make a sponge cake. If you have time to let it get slightly stale or dry it's even better, as the cake won't get soggy when topped with ice cream and coated with meringue. Then you add a nice big loaf of ice cream (bonus points if you make it yourself), shaped into a dome or whatever form you choose. The ice cream needs to be really frozen before it bakes (that's right, against all logic you're putting ice cream in an oven), so the cake and ice cream go into the freezer for a while.
Once it's good and frozen, you slather the whole thing with a soft meringue -- made from scratch, of course -- and bake the entire delicious mound for about five minutes, just long enough for the meringue to brown. An alternative to baking is to brown the tips of the meringue with a blowtorch, if you have one lying around and aren't too attached to your eyebrows.
Why doesn't the ice cream melt? The cake and the meringue insulate the ice cream just enough to keep it from turning to liquid during its brief stint in the oven.