Activate the Yeast
In a small bowl, mix warm milk, yeast, and 1 tablespoon of sugar. Let sit for 5–10 minutes until foamy.
Make the Dough
In a large mixing bowl, whisk together flour, remaining sugar, and salt.
Add egg yolks, whole egg, melted butter, vanilla, rum/brandy (if using), and the yeast mixture.
Mix until a sticky dough forms, then knead on a floured surface or with a stand mixer dough hook for 8–10 minutes until smooth and elastic.
First Rise
Place dough in a greased bowl, cover, and let rise in a warm place until doubled in size about 1 ½ hours.
Shape the Pączki
Punch down the dough and roll to about ½-inch thick.
Cut into rounds using a 3-inch cutter. Place on a floured baking sheet, cover loosely, and let rise for 30–40 minutes until puffy.
Fry
Heat oil in a deep pot to 350°F (175°C). Fry 2–3 at a time for 1–2 minutes per side until golden brown.
Remove with a slotted spoon and drain on paper towels.
Fill and Finish
Using a pastry bag with a long tip, pipe jam or custard into each doughnut.
Dust generously with powdered sugar before serving.