In a medium bowl, sift flour, baking powder, salt, nutmeg, and cinnamon. Set aside.
In a small bowl, mash the banana using the back of a fork till smooth with only a few lumps. Add the sour cream to this and fold together until just loosely dispersed. Set aside.
In the bowl of an electric mixer fitted with a paddle attachment, mix sugar and shortening together on low speed until sandy.
Add egg yolks and turn speed up to medium. Beat until light and creamy, about two minutes. Scrape down the sides of the bowl as needed.
Turn mixer down to low speed then add about a third of the dry mixture followed by a third of the sour cream mixture. Alternate adding a third each time until finished. Scrape the sides of the bowl as needed. Once finished adding all of the mixes, add in walnuts. Stir until combined but do not over mix. The dough will wet and sticky. Wrap the bowl tightly in plastic and refrigerate for at least 30 minutes or overnight.
Make the icing just before dough is finished chilling. In a small bowl, whisk together powdered sugar, maple syrup, cinnamon, salt, and half of the water. If the mixture, is thick add remaining water slowly until it reaches a smooth consistency that is as viscous as the maple syrup. Set aside.
Once the dough is chilled, begin heating a pot of oil that is at least 2 inches deep to 325° F. Line a metal cooling rack with paper towel and set another metal rack over a cookie sheet.
Scrape chilled dough out of the bowl onto a generously floured work surface. Roll the dough to be 1/2 inch thick all around using a rolling pin dusted with flour. Punch out doughnuts using a 3 inch cookie cutter ring. Use a 1 inch cookie cutter ring to punch out holes from the middle of each doughnut. Ensure your hands are dusted with flour when handling the doughnuts as it will be sticky. Use the excess dough to re-roll again and punch out more doughnut holes.
Dust off any excess flour from the doughnuts before gently placing in heated oil. Do not overcrowd the oil. Fry on the first side until it begins to float at the surface. Then flip and continue to fry for 90 seconds, the top should begin to show deep cracks. Flip again and fry for 60 more seconds or until evenly golden brown. Remove from the oil and let drain on paper towel.
While the doughnuts are still hot, dip the side with deepest cracks into the icing. Let drain on the other metal rack for 15 minutes before serving.