In your mixer bowl fitted with a dough hook, combine 2 tablespoons warm water, yeast, and 1 teaspoon sugar. Let stand for 10 minutes.
Add the butter, salt, and the egg to the yeast mixture. Stir with the dough hook. Add 2 cups of the flour and stir until incorporated with the dough hook. Add the remaining flour, 1/2 cup at a time, and knead with the dough hook. If the dough is too sticky, add 1/4 cup additional flour and knead until thoroughly incorporated. If the dough is still too sticky, carefully add more flour 1 tablespoon at a time. Continue kneading with the dough hook until the dough cleans the sides of the bowl, is satiny, doesn't stick to your fingers, and doesn't keep its shape when pinched. It should still feel slightly sticky. Too much flour will make the donuts tough.
Cover the dough with a clean dishtowel or plastic wrap and place in a warm area. Allow dough to double in size, about 1-1/2 to 2 hours.
Punch down dough and divide in half. For each half, roll the dough to about 1/2 inch thick between two sheets of parchment paper. Remove the top sheet of parchment paper and cut rounds from the dough with a round cookie cutter, cutting pieces as close together as possible, and place on a parchment-lined baking sheet. (Don't reuse the dough scraps.)
Allow to rest for 30 - 60 minutes until donuts are puffy.
Heat the vegetable oil to 350 degrees (F). Fry the donuts turning when dark golden, about 1 - 2 minutes per side. Lift from oil and allow to drain.
When cool, use a paring knife or skewer to make a small hole in one side of each donut and create a small cavity inside. Then use a piping bag fitted with a small tip, a turkey baster, or a pastry syringe to fill the cavity with the jelly of your choice. Note: Jams or thick spreads will clog the nozzle.
If desired, sift confectioners' sugar over the tops.
Feeling brave? Let party guests fill their own donuts! Fry the donuts ahead of time and remove the hot oil from the vicinity. (This is crucial for children's parties.) Set out a variety of jellies. Let each guest inject whichever flavor (or combination) they like into their donut or donuts.