1. In the bowl of an electric mixer, add the yeast, 1 tsp sugar, and the warm water and mix together with the whisk attachment. Add the egg yolks and the warm milk and continue to whisk.
2. Add 4 to 5 cups all-purpose flour, 1 tsp salt, 1 tsp vanilla extract and mix together using the dough hook attachment. Slowly add 1/2 cup of melted butter.
3. At this point, you'll probably have to add more flour. Gradually add flour, 1/2 cup at a time, until the dough is sticky but not wet and pulling away from the sides of the bowl.
4. Flour your counter and place the dough on top. Flour the dough and knead for a minute until smooth.
5. Spray a large pot with cooking spray, plop the dough in and let it rise in a warm place for an hour (or until it's doubled in size.)
6. Tear off some of the dough (about 1/3rd to 1/2), and place it on a floured surface. Sprinkle with more flour and roll out until 1-inch thick. Cut the dough into doughnuts using a 4-inch cutter and a smaller cutter for the centers (unless you're making jelly doughnuts).
7. For jelly doughnuts, take the circle of dough and add 1 tsp of jelly or jam (any flavor) into the middle. Seal with another circle of dough (make sure you pinch it together tight.)
8. Heat a shallow layer of canola oil (about 1-inch deep) in a pot until it reaches 325 degrees F (use a thermometer.) Gently and carefully add the doughnuts to the oil, being careful not to overcrowd the pan. (When you add the doughnuts, the temperature will drop, so don't add too many at once.)
9. Fry the doughnuts until golden brown on one side, then flip over. Continue to cook for a total of five minutes. Drain the doughnuts onto a plate with paper towels and then immediately toss either with sifted powdered sugar or a mixture of cinnamon and sugar and serve!