Place the yeast and 1 tsp sugar into a small bowl, and add ¼ cup lukewarm milk. Stir, cover and set aside for about 10-15 minutes.
Using a standing mixer with the dough attachment mix the flour and the rest of the sugar. Add the yolks, the yeast-milk mixture, the melted butter, the lemon zest, the vanilla extract, a few drops of rum/rum aroma, and add the milk gradually until the flour mix forms a soft dough. The dough is good when it separates from the wall of the mixing bowl.
Form a ball, lightly flour the bottom of the bowl, and place the dough back in the bowl. Cover it with a kitchen towel or plastic wrap, and put it in a warm place (I preheat my oven to 200 °F/90-95 °C for 1-2 minutes, then turn it off) for about 45-60 minutes, until it doubles the size. On a lightly floured surface roll out the dough to about ½ inch (1-1.5 cm). Cut out the doughnuts, and place them on parchment paper lined or lightly floured baking sheets.
Cover them with a kitchen towel, and let them rest in a warm place for another 20 minutes.
In a large heavy bottom skillet or dutch oven heat the oil (fill to about 1-2 inches deep). Fry the doughnuts until golden - about 1-2 minutes each side - on medium low heat, about 3-4 at once in the skillet. I usually cover the skillet for about a minute right after placing in the doughnuts, then uncover and continue to fry without it. I use long metal tongs for frying.
Once they are golden on both sides place doughnuts vertically beside each other in a paper towel lined large dish.
Sprinkle with icing sugar, and serve warm with some apricot preserves, plum butter, or any other kind of fruit spread. Bon appétit!