Stir a teaspoon of sugar in a few tablespoons of lukewarm milk and add the yeast. Leave it to rise in the warmth. Within about 15 minutes, bubbles should appear on the surface, indicating that the starter is ready to use.
Meanwhile, cream the egg yolks with the sugar and softened butter in a bowl. Use an electric mixer.
Add the remaining ingredients to the beaten egg yolk mixture: flour, the rest of the lukewarm milk, lemon zest, salt, and rum. Work thoroughly into a smooth elastic yeast dough. If too sticky, dust the dough with a spoonful of flour and work it in.
Cover the koblihy dough with a clean tea towel and let rise in a warm place for 45 minutes.
Roll out the dough on a floured worktop to a thickness of about ½ inch (1 cm). Cut out 2-2. ½ inches (5-7 cm) diameter rounds using a small glass or donut cutter. Work the remaining dough again (you should not need much flour), roll it out, and cut the rounds again.
Leave the rounds in a warm place to rise for a further 30 minutes.
Heat the oil in a saucepan over medium heat, but not too high. Put enough oil; the donuts must not touch the bottom. I used about 4 cups of sunflower oil for a 10-inches (24 cm) pan.
Place the risen koblihy top side down in the hot fat so that they swim in it and do not touch each other. If the oil is heated just right, the donuts will rise beautifully.
Fry on one side for a minute and a half, then flip and fry for another minute and a half on the second side. Remove with a metal (not plastic!) slotted spoon or colander and let the excess fat soak in on a plate lined with paper towels.
Fill the still warm donuts with a filling injector or a decorating bag with a simple tip. Stir the jam smooth beforehand, or heat it warm with a spoonful of rum so that you can push it through the tip without any hurdle.
Dust the filled koblihy with icing sugar and serve.