Take a large mixing bowl, add flour and make a dent in the middle. Place instant yeast there, sprinkle with 1 teaspoon of sugar. Pour in about ½ cups of lukewarm milk. Mix the yeast, sugar, milk, and little flour with a fork. Leave it in a warm place for 30 minutes; the yeast mixture should start bubbling.
Melt butter and let it cool for a bit. Add it to the rest of the milk in a bowl. The mixture can't be hot, only warm. Add eggs, sugar, a pinch of salt, rum, and whisk everything.
Add the mixture to the flour with proofed yeast and make the dough, either by hand with a wooden spoon or using a kitchen mixer with a proper attachment. The resulting dough should be a bit sticky, not too dense. This helps in making fluffy and pillowy vdolky.
Let the dough rise in a warm place for 45-60 minutes. The dough's volume should be twice as big as before.
Dump the risen dough on a lightly floured surface and roll it out so that it's about ½ inch thick (not too thin!). Cut the round shapes out with a flipped glass. Cover with a clean kitchen cloth and let it rise again for 20-30 minutes until puffy.
Meantime, heat frying oil in a pot over medium heat.
Before you fry a vdolek, make a shallow dump with your index finger in the middle. Put a vdolek into the hot oil carefully, with the hole facing downwards.
Fry until golden brown, then flip the vdolek and fry the second side. It's quick, about 1 minute on each side.
Remove fried vdolky from the oil carefully and place it onto a rack.
Spread the plum jam on the surface of the vdolek on the side with a hole. Sprinkle with grated tvaroh.