Preheat the oven to 375 degrees F. Grease a 9-inch springform pan and sprinkle with bread crumbs. Turn it upside down over the kitchen sink and knock gently on the bottom to remove excess crumbs.
For the dough, beat the butter with half of the sugar,the vanilla, and egg yolks, using an electric mixer.
Grind the almonds in a food processor until very fine and then combine with both flours and the baking powder.
Beat the egg whites until they stand in stiff peaks and add the remaining sugar while beating. Gently but thoroughly mix the whites with the butter and flour mixtures.
Pour the dough into the prepared pan and bake for 50 minutes, or until a tester comes out clean.
Meanwhile, prepare the filling. Pick over the cranber-ries and cook them with 1/4 cup water and the sugar until they burst. To get the required thick consistency, dissolve the cornstarch in 1/4 cup cold water and add it to the cranberries. Let it boil briefly until the cranberries turn clear again. Remove from the heat and cool.
Remove the cake from the springform pan and cool it completely on a rack. Slice it horizontally into 3 equal layers. A neat and easy way of doing this is to first use a sharp knife to make two 4-inch incisions all around the cake. For each incision, end at the same spot where you started. Then take a piece of heavy twine or dental floss the diameter of the cake plus enough to twist around your fingers. Pull the twine taut between your fingers and slide it into the incision. Using the twine like a saw, cut slowly through the cake. Spread the cranberry filling on the first and second layers and reassemble the cake.
For the frosting, combine the confectioners' sugar with the rum and 1 tablespoon water and stir until smooth. Add a few more drops of water if necessary. Spread the frosting evenly over the entire cake and let it set.