Immediately when the eggs are removed from the refrigerator, separate the whites and the yolks, placing the yolks into a medium bowl and the whites into your stand mixerbowl or another large bowl for beating. Let the whites sit on the counter until they come to room temperature, no longer than 30 minutes.
Beat the room temperature egg whites on low speed until foamy. Then, add 2 teaspoons lemon juice and beat on medium or medium high until the whites are smooth and glossy and stand straight up 'stiff peaks'. This may take upwards of 3-5 minutes, but be careful not to overmix.
Meanwhile, in a large mixing bowl whisk together flour, sugar, baking powder, and salt.
Add ripe banana to the egg yolks and mash. Mix in oil, ¾ cup cold water and dark rum.
Pour banana mixture into the dry ingredients and whisk until light and smooth, at least 1 minute.
Slowly pour banana mixture over the beaten egg whites and gently fold until just combined, being careful not to overmix.
Pour into ungreased 10" tube pan with removable bottom and legs. Bake 55 minutes or until the top slightly springs back when you press it and a toothpick inserted in the top comes out clean.
Invert pan onto legs and cool at least 45 minutes before reinverting pan. Then, carefully run a knife around the edge of the pan just to release the cake and turn onto cake plate. Remove the top in the same way.
Glaze with lemon icing (or substitite dark rum for the lemon juice for a spiced rum icing), slice with a long serrated knife and serve.
Lemon Glaze Icing
Meanwhile make the lemon glaze icing. Whisk lemon zest, remaining 1 tablespoon lemon juice and ½ cup confectioner's sugar until smooth.
Cake will keep well for at least 3 days, covered. Or, can be baked and frozen up to 3 months. Simply seal the cooled cake in plastic and freeze. Then, remove and let defrost for 2-3 hours before serving.