To make the sufganiyot, stir together the warm water, yeast, and the pinch of sugar in a large bowl. Allow the mixture to stand for a couple of minutes to allow the yeast to swell. Stir in the remaining sugar, the milk, vanilla, eggs, oil or shortening, salt, and as much of the flour as needed to create a soft dough. Knead the dough by hand or use the dough hook on a standing mixer, adding more flour as needed to form a firm dough that is smooth and elastic.
Place the dough in a greased bowl and cover the bowl tightly with plastic wrap. Let the dough rise for about 1 hour. (If not using right away, you can refrigerate the dough at this point.)
Gently deflate the dough. (If the dough is coming out of the fridge, allow it to warm up at room temperature for about 40 minutes before punching and proceeding.)
Pinch off golf ball-size pieces of dough and form them into small balls. Cover the sufganiyot with a clean tea towel and let them sit at room temperature for 20 to 30 minutes.
When you're ready to fry, heat about 4 inches of oil in a deep fryer or a heavy Dutch oven to about 385°F (196°C). Line a baking sheet with a couple layers of paper towels.
Gently press the sufganiyot to flatten them ever so slightly. To test the temperature of the oil, it's a good idea to start out frying a single doughnut. When the doughnut appears golden brown on both sides, take it out and cut it open to see if the inside is cooked through. You want to attain and maintain an oil temperature that's hot enough so it bubbles but not so hot that the doughnut browns before the center is cooked. It may take a few tries to find the right temperature and timing. When you're satisfied that the oil is at a good temperature, carefully add the sufganiyot, 3 or 4 at a time, to the hot oil and fry until the undersides are deep brown. Turn over once and finish frying the other side. The total frying time will be somewhere between 1 1/2 and 3 minutes. Lift the sufganiyot out with a slotted spoon and drain them well on paper towels.
If desired, place some granulated or confectioners' sugar in a paper bag, add a couple of doughnuts, and gently shake to coat. Use the tip of a spoon to make a small indentation in the top of each sufganiyot, much as you would with a thumbprint cookie, and spoon in a little jam or jelly.