Warm the milk on the stove or in the microwave until lukewarm (105 to 110 F).
Combine the milk with the yeast and 1 teaspoon of sugar, then cover this mixture with plastic wrap or a towel and let it sit for 10 minutes until it becomes foamy.
Stir the remaining sugar into the yeast mixture along with the eggs, yolks, salt, 5 ¼ cups flour, and vodka or rum if using.
Using a stand mixer with the dough hook attachment, knead the dough for 5-6 minutes on low speed. If dough is sticky, add remaining flour a tablespoon at a time until workable.
Slowly add the butter to the dough, a pat at a time, kneading it for 4-5 minutes until all of the butter has been added and the dough appears uniform.
Cover the dough with plastic wrap or a towel and allow it to rise for 60 to 90 minutes until it doubles in volume.
Roll out the dough on a lightly floured surface to a ½-inch thickness.
Cut the dough into 3-inch rounds using a cookie cutter or the rim of a glass.
Set the dough rounds on a parchment paper-covered and/or lightly floured surface and cover them with plastic wrap or a towel.
Let the dough rise for 30 to 45 minutes until it becomes puffy and has a marshmallow-like texture when poked.
Heat 2-3 inches of oil to 350 F over low heat.
Fry 2 to 4 doughnuts at a time, depending on the size of the pot, for 2 or 3 minutes, then flip them over and cook them until they are golden brown all over (lower the heat if they brown too quickly).
If you're dipping the doughnuts in cinnamon sugar, do so while they are still hot, then let them cool.
Once all of the doughnuts are cooked and cooled, use a piping bag or a Ziploc with a corner snipped off to fill them with jam. If using a piping bag, the tip will poke the necessary hole. If using a ziploc, you'll need to make a small incision in each doughnut to insert the filling.
Sprinkle the filled doughnuts with powdered sugar, if using.