To begin, warm the milk until lukewarm, not hot. Stir in a teaspoon of sugar and the yeast. Leave for 10 minutes until foamy. Transition to preparing the dough base.
In a large mixing bowl, combine flour, sugar, vanilla sugar, and salt. Add the activated yeast mixture, eggs, and melted butter. Mix with a wooden spoon until the dough comes together. Move to kneading.
Transfer the dough to a floured surface. Knead for 8–10 minutes until smooth and elastic. If sticky, dust lightly with flour, but avoid adding too much. Transition to first rise.
Place the dough in a lightly oiled bowl, cover with a cloth, and leave to rise in a warm place for 1–1.5 hours until doubled in size. Move to shaping.
Roll out the risen dough to about 1 cm thickness. Using a round cutter, cut circles of 7–8 cm diameter. Place on a tray, cover lightly, and let rest for 20 minutes. Transition to frying.
Pour sunflower oil into a deep pan or pot, filling to about 5 cm depth. Heat gently to 170–175°C. Test with a small piece of dough: it should rise slowly and bubble. Move to frying.
Carefully lower the dough circles into the hot oil. Fry 2–3 minutes per side until golden brown. Turn only once for even puffing. Move to draining.
Remove fried doughnuts with a slotted spoon and place on paper towels to drain excess oil. Transition to filling.
Once cooled slightly, use a piping bag fitted with a nozzle to inject plum or apricot jam into the centre of each doughnut. Move to finishing touches.
Generously dust with icing sugar. Serve warm with coffee or tea. Presentation tip: place on a platter lined with a lace napkin for a traditional touch.