These banana bread doughnuts are fried to a golden crisp, dipped in sweet glaze, and topped with chopped walnuts.
In a large bowl, whisk together flour, cake flour, baking powder, salt, and cinnamon. Set aside.
In the bowl of a stand mixer fitted with a paddle attachment, cream together brown sugar, sugar, vegetable shortening, egg, and egg yolk. Scrape down bowl as needed to ensure thorough mixing.
Add vanilla, sour cream, and mashed bananas. Mix on medium speed until well combined.
Add dry flour mixture in three additions. Mix on low speed until there are no longer any dry streaks of flour. The dough will very tacky and sticky. Use a sturdy spatula or a bowl scraper to transfer dough to a medium bowl. Cover with plastic wrap or lid and let dough chill in the fridge for at least 1 hour (overnight is best).
Attach candy thermometer to a large heavy bottomed pot (cast-iron or Dutch oven, preferred). Fill pot about 3-inches deep with shortening or oil. Set over medium heat to 350 degrees F.
Remove chilled dough from fridge. Dough will still be sticky. Transfer dough to a generously floured work surface. Flour the top the of the dough and pat (or roll) dough to a thickness of a little less than 1/2-inch. Using a floured 3 1/2-inch doughnut cutter (or round cookie cutters), stamp out doughnut rings. Transfer doughnut rings to lightly floured baking sheet.
Gather dough scrapes, knead together, and re-roll to stamp out more doughnuts.
Brush off any excess flour from the doughnut rings. Once oil comes to temperature, carefully place 2-4 doughnuts at a time into hot oil. Do not overcrowd the pot. Cook for 2 minutes on each side, until golden in color.
Drain cooked doughnuts on a wire rack over a paper towel lined baking sheet. Keep an eye on thermometer and adjust heat accordingly to maintain a constant cooking temperature.
Sift confectioners sugar into a large bowl. Stir in salt, cinnamon, corn syrup, and vanilla. Once mixture is well combined, add in hot water. Mix until smooth.
Dip warm doughnuts into glaze. Allow excess glaze to drip back into bowl. Set glazed doughnuts on wire rack and immediately sprinkle chopped walnuts on top. Allow glaze to see. Enjoy doughnuts warm or at room temperature.
A word on leftovers: Doughnuts are best enjoyed the day they are fried. Store any leftover doughnuts in a breathable box (like a cardboard bakery box or container with the lid slightly jar). The glaze needs air circulation, otherwise it will melt and make the cake doughnuts soggy.