For the fried chicken skins: Add the chicken skin and buttermilk to an airtight container, cover and refrigerate overnight. For the dough: Add the warm milk to the bowl of a stand mixer and then gently whisk in the yeast and sugar. Set aside until the yeast "blooms" at the surface of the milk and is bubbly and foamy, 5 to 10 minutes. Add the eggs followed by the flour, the butter, and lastly the salt. Use the dough hook attachment to combine the mixture on low speed; once the dough starts to come together and there aren't many dry patches left, increase the mixer speed to medium and mix until it comes together and is smooth yet still slightly sticky, about 8 minutes. Lightly coat a large bowl with nonstick pan spray and transfer the dough to the bowl. Mist the top of the dough with nonstick pan spray, then cover the bowl with plastic wrap and set the dough aside in a warm, draft-free spot until it has doubled in size, about 1 hour (depending on how warm your kitchen is). 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 (you can also pat the dough out into a 1/2-inch thick sheet or use the nonstick spray can or even a wine bottle to roll it out). Fill a small bowl with flour and dust a parchment paper-lined rimmed baking sheet with flour and set aside. Dip a 2 1/2 to 3-inch doughnut cutter into the flour, then use it to stamp out 12 to 15 circles from the dough. Place the doughnuts on the floured parchment-lined baking sheet and set aside in a warm, draft-free spot until they have nearly doubled, 45 minutes to 1 hour. Set a wire cooling rack over a paper towel-lined baking sheet. When the doughnuts are nearly ready, pour the oil into a large pot (or deep fryer) set over medium-high heat; the oil should fill the pot by about 3 inches-if it doesn't, add more. Heat the oil until it reads 325 degrees F on an instant-read thermometer. Add a few doughnuts to the hot oil, taking care not to overcrowd the pot (otherwise the doughnuts will stick together and not cook evenly). Cook until the doughnuts are golden, about 1 to 2 minutes, then use extra-long chopsticks, a fork, or a slotted spoon to gently turn the doughnuts over. Fry on the other side until golden, about 1 to 2 minutes longer. Transfer the doughnuts to the wire rack to drain and cool, then repeat with the remaining doughnuts, making sure the temperature of the oil comes back to 325 degrees F between batches. For the fried chicken skins: Increase the heat under the pot with the oil to medium-high and heat to 350 degrees F. Line a plate with paper towels and set aside. Add the flour, pepper and salt to a brown paper bag, fold the top down to seal and shake to combine the ingredients. Remove the chicken skins from the buttermilk letting the excess liquid drain off, then add the skins to the bag with the flour mixture. Fold down the top and shake the bag to coat the chicken skins with the seasoned flour. Remove the skins from the bag shaking off the excess flour mixture, then add them to the hot oil. Fry until the skins are browned and crispy, 3 to 5 minutes. Transfer the fried chicken skins to the paper towel-lined plate and season with a few pinches of salt, then, once cool enough to handle, break into large shards.
For the maple glaze: Sift the confectioners' sugar through a fine-mesh sieve and into a large bowl. Add the maple syrup, milk, vanilla and salt and whisk to combine. (The glaze should be on the thick side since the warm doughnut will thin it slightly; if it seems too thick, however, add more maple syrup or milk; if it gets too thin, add a little more sifted confectioners' sugar.) Take a warm doughnut and dip once side into the maple glaze. Turn the doughnut over and add a few pieces of chicken skin to the top. Repeat with the remaining doughnuts, glaze and chicken skins. Let the glaze set up for a few minutes before serving.