Place the flour, yeast and sugar in a stand mixer bowl. Attach the dough hook and mix on low speed. Pour in the milk and add the butter. Add the eggs and vanilla, then the salt. Knead until the dough is smooth and sticky, about 10 minutes.
Transfer the dough to a greased bowl, cover and let rise for 1 hour at room temperature or overnight in the refrigerator.
Turn the dough onto a floured surface. Divide into 1.8-ounce (50–gram) pieces and roll into tight balls. Set each ball on a small square of parchment, cover with a towel and let rise until doubled.
Heat the oil to 340°F/170°C.
Lower the donuts into the oil using the parchment. Fry for about 2 minutes per side, until golden. Drain on a wire rack. Cool slightly, fill with jam using a piping bag and dust with powdered sugar.