For the doughnuts: Whisk together the cake flour, baking powder, cinnamon, salt and nutmeg in a large bowl until well combined. In the bowl of a stand mixer fitted with the paddle attachment, cream the sugar and shortening together on medium speed until well combined (the mixture will look sandy), 1 to 2 minutes. Add the egg yolks one at a time, waiting until each is incorporated before adding the next, then add the maple extract. Increase the speed to medium and beat the mixture until light and fluffy, 4 to 5 minutes. Reduce the speed to medium-low and add the cake flour mixture alternately with the sour cream in 3 additions, starting and ending with the dry ingredients. Continue to mix the dough until it is stiff and sticky, 1 to 2 minutes. Cover the bowl with plastic wrap and refrigerate at least 1 hour or overnight.
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 circles out from the dough, then 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 15 doughnuts.
Line a rimmed baking sheet with paper towels and set a wire cooling rack on top. Pour the oil into a large saucepan 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 335 degrees F on an instant-read thermometer. Add 2 to 3 doughnuts to the hot oil and, once the last doughnut rises to the surface, cook without turning the doughnuts for 30 seconds. Turn the doughnuts over and fry for 1 1/2 minutes, then turn the doughnut over and fry until they are golden-brown and have deep golden-brown cracks running along the surface, about an additional 1 minute and 15 seconds (you can remove 1 doughnut from the oil and carefully slice it in half to make sure the inside is cooked through). Transfer to the wire rack to cool and repeat with the remaining doughnuts, making sure the temperature of the oil returns to 350 degrees F between batches. Keep the oil at 350 degrees F for the chicken and pickles.
For the glaze: Whisk together the confectioners' sugar, melted butter, maple syrup and hot sauce in a medium bowl until smooth. Gradually whisk in the hot water a little at a time until the glaze is runny but not too thin. While the doughnuts are still warm, dip one side of each doughnut in the glaze, then return it to the cooling rack.
Drain the chicken pieces and add them to the other bowl with the flour mixture, tossing them to evenly coat. Shake off the excess then add the chicken pieces to the hot oil, frying until the chicken pieces are golden-brown and crispy, about 3 minutes. Transfer the fried chicken to the paper towel-lined plate and immediately sprinkle with kosher salt. Top each doughnut with a few fried pickles and a few pieces of fried chicken and serve.