In a large bowl or the bowl of your stand mixer, weigh out and add all-purpose flour, sugar, salt, yeast, egg yolks, brandy (or rum), milk, melted butter and vanilla extract.
If you are kneading by hand, stir the mixture with a spoon until it all comes together and then knead for 15 minutes until you have a smooth dough. If you are using a stand mixer, knead for 8 to 15 minutes on medium speed using the dough hook.
Move the dough into a lightly oiled bowl and leave in a warm spot in your kitchen for 1.5 to 2 hours to rise. After this initial rise, I like to put the dough in the fridge overnight. This isn't required, but it helps with the 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 2 hours while the paczki dough rests.
After the dough has risen, gently deflate it and move to a lightly greased or floured surface. Using a rolling pin or your hands, form the dough into a rough rectangle and roll out or flatten to about 1/4 to 1/2 of an inch in height.
Using a round cookie or biscuit cutter (or even a sharp-edged glass or cup) cut out 2.5-to-3-inch circles. Place each circle onto a sheet pan lined with parchment and cover with plastic wrap or a second inverted sheet pan and allow the dough to rise again for 30 minutes. Do not discard any extra dough after cutting into circles. Push leftover dough together and re-roll and re-cut until you have used all the dough. 1/4 inch thick seems thin, but these puff up big in the frying process.
In a wide pan or dutch oven, bring 1.5 to 2 inches of peanut or vegetable oil up to 350 degrees F (175 C).
Fry 2 or 3 paczek at a time, for about a minute per side until golden brown. Flip and fry on the second side for another minute.
Remove each paczek to a cooling rack set inside a sheet pan to cool. If you want the top covered in powdered sugar, you should sprinkle that on when the paczek is hot.
Using a piping bag or zip-top bag that has one corner cut off, insert and fill paczki with jam or custard. If you do not have a piping bag, you can simply slice each paczek and spread your filling of choice prior to eating. When you are piping filling, I find it easier to poke a hole initially with a knife and use that knife to slash back and forth to make a big hole inside your paczek.
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.
Slice a paczek carefully and add one fried chicken thigh between the slices. It's easier if you slice below or above the level of the hole where the filling was piped in. If you already sliced your paczek just open it and add the chicken. Serve and enjoy.
If you are not serving immediately, I suggest that you do not coat all of your paczek with powdered sugar because it will eventually dissolve into the top of the doughnuts. You can add powdered sugar later by simply adding a drop or two of oil to the top of a doughnut and spreading the oil around with a brush or even your clean finger. Then the confectioner's sugar will have something to stick to. You can do this just prior to serving for a prettier paczek.
If you do not want to fry all of your paczek at one time, you can freeze them just prior to the frying step. Add them to a sheet pan and freeze for one hour until they are firm and then you can remove them from the sheet pan and freeze for up to three months in a freezer ready zip-top bag. Then when you want to fry them, you need to leave them on the counter for 1 hour prior to frying to thaw. After that one hour you can follow the rest of the frying instructions.