Mix wet ingredients: In a medium size bowl combine the eggs, milk, butter, sugar, salt and vanilla extract. Add the yeast and mix well.
Make dough: Add the flour to the bowl of your mixer, then pour in the milk/egg mixture. Using the dough hook, mix well until well incorporated. You will know that the dough is done when it comes clean from the sides of the bowl. The dough will be a soft dough.
Prepare dough and let rise: Turn over the dough onto a floured surface and knead for a couple minutes, then place the dough in a bowl that's sprayed with cooking spray. I also like to spray the dough with some cooking spray so that it doesn't dry out. Cover with plastic wrap or a clean kitchen towel and let rise for a couple hours. The dough should double in size.
Roll dough and cut rounds: Roll out the dough on a floured surface so that's it's about 1/4 inch in thickness. Cut out rounds using a glass or a cookie cutter. Repeat with the remaining dough. Place the donuts on a baking sheet lined with parchment paper and let them rise again for about 30 minutes until doubled in size.
Fry donuts: Add about 3 inches of oil to a deep frying pan, I used a Dutch Oven. Before frying the donuts make sure the oil is hot, it needs to be 350 F degrees. Fry the donuts, about 5 or 6 at a time until golden brown on both sides. Make sure you turn them on both sides to fry them evenly.
Cool: Transfer the donuts to a large bowl lined with paper towels to drain some of the excess oil. Allow them to cool until you are able to handle them.
Fill: Using a piping bag and a filing tip, pipe your favorite filling into the sides of the paczki.
Roll the donuts in powdered or granulated sugar then serve.