In the bowl of a stand mixer, add warm milk. Stir in yeast and a pinch of sugar. Let stand until the mixture becomes bubbly, about 5 minutes.
Add 2 cups of flour, and stir by hand until batter is smooth. Cover the bowl with plastic wrap, and store in a warm place for 30 minutes to allow mixture to rise.
In a separate bowl, whisk the egg and egg yolks until they are frothy.
Add sugar, vanilla extract and salt to the egg mixture, and whisk until smooth.
Attach a dough hook to the stand mixer. Add egg mixture to the dough and mix on low until combined. Add melted butter and mix on low until combined.
Gradually add up to 3-4 more cups of flour until the loose dough just comes together. Transfer the dough to a lightly greased bowl. Store in a warm place for 30 minutes.
Turn the dough onto a floured surface. Sprinkle with flour and roll to a 1/2-inch thickness.
Line two baking sheets with parchment paper. Use a round cutter to cut as many 3-inch rounds of dough as possible, gathering and re-rolling the remaining dough as needed.
Cover the baking sheets loosely. Store in a warm place for 30 minutes.
In a large, heavy skillet, heat 1 1/2 inches of oil to 350 degrees F. Using a slotted spoon, gently lower 4-5 paczki into the oil. Fry the bottom of each paczki until golden brown before flipping to brown the other side. Transfer the packzi to a paper towel-lined baking sheet. Allow to cool.
Use a filling tip to inject fruit preserves into the center of each paczki. Roll the finished paczki into the sugars to lightly coat.
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