Traditional Polish Pączki (Doughnuts)

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1. 1Step 1: In a large bowl, whisk together the flour, sugar, yeast, and salt until combined.
2. 2Step 2: Heat 1 cup of milk to between 110 F and 115 F. Add the warm milk, whole egg, egg yolks, melted butter, and rum together, then add it to the flour mixture. Mix until a sticky dough comes together.
3. 3Step 3: Knead the dough with floured hands, until the dough becomes smooth and elastic, about 6 to 8 minutes. Once kneaded, place the dough in a clean greased bowl, cover, and let rise in a warm place until it doubles in size, about 1 to 2 hours.
4. 4Step 4: Once the dough has doubled, place the dough on a lightly-floured surface, and roll it out until it's 1-inch thick.
5. 5Step 5: Use a 3-inch biscuit cutter to cut out rounds, and place them on baking sheets lined with parchment paper. Cover the sheets, and let the dough rise in a warm place, until the dough doubles in size again, about 1 hour.
6. 6Step 6: Pour enough oil into a large dutch oven until it is at least 3 inches deep. Heat oil to 345 F.
7. 7Step 7: In batches of about 4 or 5 at a time, fry donuts in the hot oil for 2 to 3 minutes per side, then place them on a paper towel-lined baking sheet to remove excess oil.
8. 8Step 8: In a large bowl, add the pudding mix of your choice, and the remaining 2 cups of milk, and whisk together until combined. Let sit until the pudding thickens.
9. 9Step 9: Once the donuts are fully cool, about 1 hour, use a knife to cut a hole in the side of each one. Put the pudding into a piping bag, and pipe the pudding into the center of the donuts, or use a spoon to scoop the filling in.
10. 10Step 10: Sprinkle the donuts with powdered sugar, if desired.