Adjust rack to one-third up from oven bottom. Have ready an assembled, two-piece, 10" by 4" tube pan. Have ready a baking sheet to place under the tube pan as the cake bakes.
Separate eggs (it's easiest to do this while they're still refrigerator-cold). Place the yolks in a bowl of about three-quart capacity (or the small bowl of an electric stand mixer). Place the whites in the large bowl of an electric mixer, which must be perfectly grease-free and of at least a 4-1/2 quart capacity. Cover both bowls and leave the whites and yolks to warm up to room temperature (about an hour).
Meanwhile, divide sugar by measuring out 1/2 cup; place near bowl of egg yolks. Measure out remaining 1 cup and place near bowl of egg whites. Into a medium bowl, sift together the potato starch and unsweetened cocoa powder. With a fork, blend well, trying to get out as many lumps as possible (potato starch tends to clump--don't worry too much about it). Assemble other ingredients and a food processor (fitted with steel blade).
When whites and yolks are about at room temperature, preheat oven to 350 degrees F. With powerful, hand-held electric mixer (or in small bowl of electric mixer fitted with whisk beater), beat egg yolks at low speed to combine. Increase speed to high; beat for 3 full minutes, or untol yolks are pale and have increased slightly in volume. At low speed, add vanilla, then gradually incorporate 1/2 cup sugar. When sugar has been added, scrape bottom and sides of bowl with rubber spatula; increase speed to high again and beat for 2 full minutes. Add oil, but do not mix in.
Working quickly, peel bananas and cut off any brown or soft spots. Slice, then process in food processor at high speed just until smooth. Measure out 1 cup of puree and add to egg yolk mixture (note: bananas can be mashed with a fork, put through a potato ricer, etc. instead of being put into a food procesor, but the resulting puree should be smooth--not chunky--and you'll need 1 cup of it). At medium speed, beat in oil and banana puree just until combined. At lowest speed, add about one-third of sifted dry ingredients and half of water; beat just until combined. Scrape bottom and sides of bowl with rubber spatula. Add half of remaining dry ingredients and all of remaining water; beat at low speed till mixed. Add remaining dry ingredients and rum, and beat at low speed just until combined (this mixture will be thick).
To bowl of egg whites, add salt and lemon juice. With clean whisk beater(s), beat at high speed until increased in volume and very foamy. Decrease speed to low; add about 2 Tbsp. of the 1 cup of sugar. Increase speed to high to incorporate sugar thoroughly. Repeat this process, decreasing mixer speed to add sugar, then increasing speed to high to incorporate each addition, until all sugar has been added, beating for about 15 seconds between each addition. When all sugar is incorporated, beat meringue at high speed just until stiff peaks form. Remove from mixer.
Using a large rubber spatula, stir a large spoonful of the meringue into the chocolate batter to lighten it. Add another large spoonful of meringue and gently fold it in (not too thoroughly). Add about one-third of remaining meringue; again, gently fold in, not too thoroughly. Pour this lightened choclate batter gently over the remaining meringue, and gently, carefully, and quickly fold the two mixtures together just until all is combined and no white streaks show. Turn into pan, which will be 2/3 to 3/4 full. Place pan on baking sheet, then immediately place in preheated oven.
Bake 45 to 55 minutes. While cake bakes, get the long-necked bottle ready. I turn the baking sheet back-to-front once, very gently, after about 35 minutes of baking, but I don't open the oven door before that. This cake will rise above the top of the pan and develop deep cracks in its top surface (they won't show when you serve it). Cake is done when top springs back after being touched. Do not overbake.
Immediately after cake is removed from oven, gently turn it upside down so that it is suspended on the neck of the bottle. Be careful!! The cake pan will probably hang at a slight angle, and it should be about a foot off the table to allow for air circulation (I place my bottle on a cooling rack, which elevates the cake a bit more). Cool cake in a draft-free area; it will shrink a bit in volume while cooling. Allow to cool completely before removing from pan; this may take as long as several hours.
When cake is completely cooled, carefully turn the pan right side up (a bit of liquid may have exuded from the cake onto the bottle--OK). To turn the cake out of the pan, loosen it from the sides first, using a stiff, thin-bladed spatula, preferably one made of plastic so it won't scratch your pan. Do not use a sawing motion in doing this. Once the sides are loosened, lift up the tube piece, and the cake should come with it. Now, loosen the cake from the pan bottom with the same spatula. Carefully and gently invert the cooled cake onto a large serving plate.
Serve the cake now, store airtight at room temperature for a day or two, or freeze for longer storage. Just before serving, sift a light dusting of confectioners' sugar onto the top and sides of the cake. Cut with a large, sharp, serrated knife, using a gentle sawing motion so you do not squash this delicate dessert.