In a large mixer bowl, whisk the yeast with the water and let stand two to three minutes.

Add the milk or water and oil or shortening. Let the mixture cool a bit and then add the sugar, salt, eggs and yolks. Stir well and pour into the yeast mixture.

Add the flour and stir on low speed to form a soft dough. Increase the mixer speed to medium and knead for five to eight minutes, adding a bit of flour as needed to form a firmer dough. When the dough is smooth and elastic, remove the dough hook, spray the dough with nonstick cooking spray and cover the bowl with a large plastic bag. Let the dough rise until almost double in size or do a cold rise overnight in the refrigerator.

Punch down the dough and pinch off pieces to form small, golf ball-sized balls. Place them on a cookie sheet or plate, cover and let rest for 15 minutes.

In a deep fryer or heavy Dutch oven, heat four inches of oil to 375 degrees Fahrenheit. Add the doughnuts a few at a time and fry until the undersides are deep brown. Gently turn over once and finish frying the other side. Lift the doughnuts out using a slotted spoon and drain on paper towels.

Slit the doughnuts open two-thirds of the way. Pipe or spoon in a tablespoon or so of jelly and gently reseal the doughnuts. Sprinkle with granulated sugar, if desired. Serve at once.