For the banana cake doughnuts: Add the cake flour, bread flour, baking powder, baking soda and cinnamon to a fine-mesh sieve and sift over a large bowl. Whisk the bananas and buttermilk together in a small bowl. To the bowl of a stand mixer add the shortening, sugar, salt and vanilla and using the paddle attachment, cream the mixture on medium speed until smooth and creamy, 1 to 2 minutes. Beat in the whole egg, then add and the egg yolk and beat until smooth. Alternate adding the flour mixture and banana mixture in 3 additions, beginning and ending with the dry ingredients and mixing just until combined. Cover the bowl with plastic wrap and set aside for 10 minutes.
Generously coat the work surface with flour and turn the dough out onto the floured surface. Dust the top of the dough with more flour, then roll the dough into a 1/2-inch thick sheet. Use a 3-inch donut cutter to stamp out circles from the dough. Place the circles on a parchment paper-lined baking sheet. Bring the scraps together, re-roll them into a 1/2-inch thick sheet, and stamp as many circles from the dough as possible (discard the remaining scraps); you should get about 12 donuts.
Line a rimmed baking sheet with paper towels and set a wire cooling rack on top. Pour the oil into a large heavy-bottomed frying pan set over medium-high heat; the oil should fill the pot by about 2 inches-if it doesn't, add more. Heat the oil until it reads 375 degrees F on an instant-read thermometer. Add 2 to 3 doughnuts to the hot oil and cook until the doughnuts are golden brown, about 1 1/2 minutes on each side. Transfer to the cooling rack to drain and cool and repeat with the remaining doughnuts, making sure the temperature of the oil returns to 375 degrees F between batches.
For the bananas foster: Melt the butter in a large skillet set over medium heat. Add in the brown sugar, cinnamon and the salt stirring until the mixture starts to bubble. Add the bananas and gently stir to coat in the sugar mixture. Continue to cook the bananas until they are warmed through, 1 to 2 minutes. Use a slotted spoon to transfer the bananas to a medium plate. Stir the rum and banana liqueur to the skillet and cook until combined, then pour the sauce into a heat-safe mixing bowl. Whisk in 1 cup of confectioners' sugar, and once well combined, begin adding more confectioners' sugar 1 tablespoon at a time, until the sauce is a medium-thick consistency good for drizzling.
For serving: Slice the doughnuts three-quarters of the way through horizontally. Stuff the doughnut with a few banana slices. Drizzle the sauce over the top and serve generously dusted with confectioners' sugar.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.