Add the lukewarm water, yeast, sugar, flour, oil, eggs, vanilla, and salt to your mixer bowl. Mix on low speed for about 15 minutes. The dough will be very sticky, and that is exactly what you want for soft donuts.
Place the dough in a large, lightly oiled bowl. Cover it with plastic wrap and a tea towel, then leave it somewhere warm. It should double in size in about 2 hours.
Line a baking sheet with parchment paper. Dust the parchment and your work surface with plenty of flour. This stops the sticky dough from clinging to everything.
Turn the dough onto the floured surface and pat it into a ½ inch thick round. Dip your cutter in flour and cut out as many circles as you can. Re-roll the scraps and cut again so nothing goes to waste.
Set the circles on the prepared sheet and dust them lightly with flour. Cover with a dry tea towel or plastic wrap. Let them puff up for about 1 hour.
Pour 2 to 3 inches of oil into your deep pot and clip on the candy thermometer. Warm it over medium heat until it reaches 325°F (165°C). Keep a close eye on it and never walk away from hot oil.
Line another baking sheet with paper towels and set a wire rack on top. Place it right next to the stove. Having it ready means you can move the hot donuts quickly.
Lower 3 to 4 donuts into the oil at a time. Fry for 2 to 3 minutes until golden, then flip carefully and fry for 1 to 2 minutes more. Adjust the heat as needed to hold the temperature steady.
Lift each donut out with a slotted spoon and set it on the wire rack. Repeat with the rest of the batch. Let them cool before you fill them.
Poke a small hole in the side or top of each cooled donut with a straw or skewer. Pipe in a little strained jam. Finish with a generous dusting of powdered sugar.