Pour warm milk into bowl of a stand mixer. Stir in the yeast and a pinch of granulated sugar. Let stand for 5 to 10 minutes, or until it has become bubbly.
Add 2 cups of flour to the mixture and stir with a wooden spoon until a smooth batter forms. Cover with plastic wrap and set in a warm spot for 30 minutes. The mixture should have risen and be very bubbly.
In a medium bowl, whisk the egg and egg yolks until pale yellow and frothy, about 3 minutes. Add the sugar, vanilla extract and salt, and whisk until combined and smooth.
Attach the dough hook to the mixer, add the egg mixture to the dough and mix on medium-low speed until mostly combined. Add the melted butter and mix to combine. Gradually add 3 more cups of flour to the mixture and continue to knead until a very soft dough comes together. (It will not clean the sides of the bowl or form a ball; it will be rather slack and a bit sticky.) If necessary, add up to another 1 cup of flour, a spoonful at a time, until the dough forms.
Transfer the dough to a lightly greased bowl, cover with plastic wrap, and set in a warm spot until it has doubled in size.
Remove the dough from the bowl and turn out onto a floured work surface. With your fingers, push down the dough into an even layer. Sprinkle flour on the dough and roll it out to ½-inch thickness. If the dough doesn't hold its shape and springs back, cover with a damp towel and let rest for a few minutes and try again.
Use a 3-inch biscuit cutter to cut out rounds of dough. Transfer the dough rounds to parchment-lined baking sheets. Gather scraps of dough and again roll out and cut until you have used up all of the dough. Cover the baking sheets loosely with plastic wrap and place in a warm, draft-free spot until almost doubled in size, about 30 minutes.
Meanwhile, heat at least 1½ inches of oil in a heavy-bottomed pot or deep skillet (I used a 12-inch cast iron skillet) over medium heat to 350 degrees F. Carefully lower about six paczki into the oil at a time (be sure not to over-crowd the pan) and fry until the bottom is golden brown. Carefully turn them over and continue to fry until the other side is golden brown. Use a spider strainer or slotted spoon to remove them to a paper towel-lined baking sheet to drain. Allow the oil to come back to temperature, then repeat until all of the paczki have been fried.
Allow the paczki to cool until you are able to handle them easily. Using a filling tip, pipe fruit preserves into the sides of the paczki, then roll in sugar. The paczki are best the same day they are made, but can be stored in an airtight container at room temperature for up to 2 days.