Combine the yeast, warm water, and 1 tablespoon of the sugar in a bowl. Let it sit for 5-10 minute until foamy to proof. When bubbly, add eggs, butter, salt, and vanilla.
In a large mixing bowl, combine the remaining sugar, warm milk, butter, egg, egg yolks, vanilla, and salt. Add the proofed yeast and 2 cups of the flour, then beat using the paddle attachment for 1-2 minutes until smooth.
Switch to the dough hook and continue to add the remaining flour, 1 cup at a time, kneading between additions. Once the flour has been added, knead on medium-low speed for about 5 minutes until smooth. The dough will be sticky and soft, but should start to clean off the side of the bowl. You can also knead by hand for 7-9 minutes on a lightly floured surface instead. If the dough is really sticking to the bowl, you can add a little more flour, ½ cup at a time, until the dough is workable. But for the lightest paczki, try not to add too much flour.
Transfer the paczki dough to a clean, lightly greased bowl. Cover with plastic wrap and let rise in a warm place until doubled in size, about 1-2 hours or in the fridge overnight.
Turn out dough onto a lightly floured surface and roll out to about ½-inch thick. Use a 3- to 3 ½-inch round biscuit or donut cutter (or cup or lid about that size) to cut out rounds, then place them on baking sheets lined with parchment paper with space between them to rise. You may need to dip the cutter into flour between each cut so it doesn't stick to the dough.
Cover the circles of dough loosely with plastic wrap and let them rise for 30-60 minutes until puffy and doubled in size. Or place them in the fridge for a slow proof overnight so they are ready to fry in the morning.
Heat about 3 inches of oil over medium heat in a large cast iron skillet to 325 to 350°F. Fry 3-5 donuts at a time for 2-3 minutes on one side, then flip and fry for another 1-2 minutes on the second side until golden brown with a pale stripe around the middle.
Transfer the fried paczki to a paper towel-lined baking sheet with a wire slotted spoon to drain excess oil. Toss in granulated sugar while still hot. Or let cool completely, then roll or dust in powdered sugar.
Fit a piping bag fitted with an open piping tip and fill with your filling of choice. Stick the piping tip into the side of a donut and squeeze until the filling starts to pool at the top of the donut. If you have a hard time getting your piping bag into the side of your paczki you can make a slit with a knife to get things started.
Serve immediately for best results, or store upright at room temperature for up to 2 days.