Weigh all ingredients into the bowl of your stand mixer. Knead in your stand mixer on medium for 15 to 20 minutes. If you do not have a stand mixer, you need to knead by hand for at least 20 minutes until the dough becomes smooth.
During the 15 to 20 minutes of kneading, the dough starts off sticky but after 15 minutes it will be smooth, and the dough should start pulling off the inside of the mixing bowl.
After the dough is smooth, add to a lightly oiled bowl, cover, and let the dough rise for 1 hour in a warm place.
After the dough has risen for an hour and nearly doubled in size, place it in the refrigerator overnight or for at least 8 hours. This step is to cool the dough, so it is easier to manipulate and handle during the cutting and shaping process.
Buttermilk marinade: add salt, pepper cayenne pepper, and paprika to buttermilk (add hot sauce if you want extra spice). and whisk to combine. Put chicken in a zip-top bag or bowl with a lid. Pour buttermilk marinade over the chicken and store in the fridge overnight or at least 8 hours while the doughnut dough rests.
Once the dough is cooled, place it on a counter and roll it out to 1/2" thickness. Use two circle cutters to punch out 3 to 4-inch donuts. I use a 3 1/4-inch large cutter.
Place each cut doughnut on its own 5-inch square of parchment that has been lightly oiled to prevent sticking.
Once the dough is on its own parchment paper, cut 1 to 1.5-inch circles in the middle. I use a 1 1/4-inch small cutter to cut out the holes.
Allow the doughnuts to rise and proof again for 15 to 30 minutes.
In a large pot bring 2 quarts of peanut oil or frying oil up to 360 degrees F (182 C).
Fry doughnuts in batches of 3 to 4 for 3 to 4 minutes flipping halfway through the cooking process. If your oil stays near 360 F you should just need to cook until both sides are a nice golden-brown color. The inside should be cooked fully at this point.
As each of your doughnuts is cooked, lay them to rest and cool off on a paper towel-lined cooling rack or just paper towels on a sheet pan. Allow the doughnuts to cool for at least five minutes while you make the glaze.
Add 1 1/2 cups confectioners' or powdered sugar to a large bowl. Add vanilla extract. In your microwave or on a pot on the stove, bring the milk up to a lukewarm temperature and then add it to the sugar and whisk to combine fully.
Glaze each doughnut one at a time by dipping it into the glaze and flipping it over once or twice. Move the glazed doughnuts to a cooling rack over a sheet pan to rest. The glaze will drip off so placing it over a sheet pan will minimize cleanup.
Once the doughnuts have rested for a minute or two after glazing, they are ready to be turned into sandwiches or eaten plain.
Seasoned flour and frying: make seasoned flour by combining flour with all of the spices, salt, and pepper in a bowl or pan that is large enough for a piece of chicken to lay flat in the flour.
Remove one piece of chicken at a time from the marinade and dredge in the seasoned flour mixture until thoroughly coated on all sides. Place the fully coated chicken onto a piece of parchment or a sheet pan rack to rest.
Heat 2 inches of peanut oil in a skillet to 350 degrees F (175 degrees C).
Fry each piece of chicken for 6 to 7 minutes or until it reaches 165 degrees internal temperature. You'll probably want to fry for 3 minutes on the first side and then flip it to make sure you're getting the level of browning that you want. Continue cooking on the other side.
After frying, place the finished chicken on a cooling rack over paper towels to drain some of the oil away.
Add one fried chicken thigh between two doughnuts and serve. For a smaller sized sandwich, you can slice a doughnut in half and use the two halves as one bun.
Sandwich options would be to add honey mustard or a drizzle of your favorite hot sauce on top of the chicken.