In a heavy saucepan combine the sugar, butter, salt and water. Bring to a boil.
Remove from heat and gradually add flour.
Put pan over a low heat and stir until it forms a ball. This really varies in time, but typically about 3 minutes.
Transfer dough to a medium bowl and then using a hand mixer, beat in eggs one at a time. Be sure to completely incorporate each egg, before adding the next.
Beat until batter it is thick and smooth.
At this point you can fry immediately or cover and refrigerate, overnight.
Pour enough oil into a large pan, I used a dutch oven, to give you a depth of about 2 inches. This is a lot of oil. You can save the oil for another cooking purpose. I'm going to use mine to fry latkes.
Bring oil to a temp of 375°, using a candy or fry thermometer. It is REALLY important to maintain the 375° temperature. Your donuts will be brown on the outside and doughy on the inside if not.
I use a small ice cream scoop but you can use tablespooons to form balls and then drop carefully into oil.
Using a "spider" strainer, turn them once or twice so they cook evenly. They should cook in about 5 minutes.
Drain on paper towels and sprinkle with powdered sugar or a cinnamon/sugar mix, while they are still hot. This will make the sugar stick.
You can serve as is or you serve with dipping sauces like chocolate or caramel.
If you want to fill them, make an X with a small knife that goes almost all the way through. Using a round tip and pastry bag, slightly fill each donut, once they are cool. I used Nutella and strawberry jam.