In a small bowl, sprinkle the yeast over the lukewarm milk. Stir briefly and allow to sit at room temperature until frothy, about 10 minutes.
In the bowl of a stand mixer fitted with a dough hook or a large bowl, combine the flour, powdered sugar, lemon zest, and salt.
Mix in the milk with frothy yeast, egg yolks, rum, vanilla extract, and butter until a soft dough forms.
Knead on a lightly floured surface just until smooth. Place in a large bowl, cover, and allow to rest at room temperature until doubled, 1-2 hours.
On a lightly floured surface, roll the dough into a sheet about 3/4 inch (2 centimeters) thick.
Cut out circles (do not twist) with a 3 inch (7.5 centimeter) circular cutter. Bring together the scraps of dough and repeat to get as many doughnuts as possible.
Place the doughnuts 2 inches (5 centimeters) apart, cover with a towel, and allow to rise at room temperature until puffy, 30-60 minutes.
Pour 2 inches (5 centimeters) of vegetable oil in a large saucepan with a lid and heat to 340˚F (170˚C), no higher.
Once the oil is heated, press your thumb down into the center a puffed doughnut, pressing all the way down but taking care to not tear through the dough.
Repeat with 3-4 more doughnuts, depending on the size of your pan, then gently place them, indentation side down, into the hot oil, taking care to not overcrowd. Immediately cover the pot with the lid.
Fry until lightly golden on the bottom, about 2 minutes, then remove the lid and flip to fry uncovered until golden on the other side.
Transfer to a towel-lined platter, indentation-side down so no oil gets trapped and soaks into the dough, and repeat with remaining doughnuts, keeping the temperature no higher than 340˚F (170˚C).
Serve the Farsangi Fánk warm with a dusting of powdered sugar and a spoonful of apricot jam in the center.