Using a large bowl, mix ¼ cup sugar, yeast, salt and 2 cups flour. Heat the milk, shortening, and water to 105℉ to 110℉ in a small saucepan. Add the milk mixture to the dry ingredients and beat on medium speed for 3-5 minutes.
Add the egg; beat on high 2 minutes. Stir in enough remaining flour to form a soft, sticky dough.
Turn the dough out onto a floured surface and knead until smooth and elastic, about 6-8 minutes.
Place in a greased bowl, turning once to grease the top. Cover and let rise in a warm place until doubled, about 1 hour.
Punch the dough down. Turn the dough out onto a lightly floured surface again and roll out to a ½ inch thickness.
Cut the dough out with a floured 3 inch round biscuit or cookie cutter. Place 2 inches apart on greased baking sheets. Cover and let rise in a warm place until nearly doubled, about 1 hour.
Heat the vegetable oil to 350° (use an electric skillet, deep fryer, or a Dutch oven pot).
When hot, fry the doughnut rounds, a few at a time, until golden brown, about 2-3 minutes on each side. Transfer to a wire rack set over a rimmed baking sheet and allow to cool slightly.
Fill a pastry bag fitted with a filling tip with the jam. Insert the filling tip into the side of a doughnut about halfway through and pipe in about a tablespoon of jam into the center and pipe in about a tablespoon of jam. Return to the wire rack and repeat until all the doughnuts are filled.
Generously sprinkle the powdered sugar on top of the baking sheet of filled paczki doughnuts.