In the bowl of a stand mixer fitted with a dough hook, add the warm milk, warm melted butter, salt and vanilla. Add the sugar and yeast, crack in the egg, and sift in the flour.
Knead the dough, shape into a ball, and let rise in a warm spot for 1.5-2 hours until well doubled.
Turn the dough out onto an oiled surface and roll into a 1 cm thick sheet. Cut out donut shapes with a cutter and a shot glass for the holes.
Transfer the shaped donuts to a parchment-lined tray and let them proof.
Pour vegetable oil into a small heavy-bottomed saucepan to a depth of no more than 1.5 cm, heat it, and fry the donuts one at a time on both sides. Place on paper towels and let cool completely.
Split the chocolate into 70% and 30% portions. Melt the larger portion over a bain-marie or in short bursts in the microwave, stirring every 5-7 seconds, then stir in the remaining 30% until smooth.
Dip the donuts in the glaze, sprinkle with sprinkles right away, and let the glaze set.