Take a large mixing bowl, add 4 cups all-purpose flour and make a dent in the middle. Place 2 ½ teaspoon active dry yeast, 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 1 stick unsalted butter and let it cool for a bit. Add it to the rest of the warm milk in a bowl. The mixture can't be hot, only warm. Add 2 egg yolks, ⅓ cup granulated sugar, ¼ teaspoon salt, 2 Tablesspoons rum, and whisk everything.
Add the mixture and the proofed yeast to the flour to make the dough, either by hand with a wooden spoon or using a kitchen mixer with a dough hook attachment. The resulting dough should be slightly sticky, not too dense, which helps in making fluffy and pillowy vdolky.
Let the dough rise in a warm place for 60 minutes, or until it has doubled in size.
Turn the raised dough onto a lightly floured surface and roll it out to about ½ inch thick pancake. Use an upside-down glass to cut out round shapes. Cover with a clean kitchen towel and let them rise again for 20-30 minutes until puffy.
Meanwhile, heat the frying oil in a saucepan over medium heat. If you have an instant-read thermometer, the oil should reach 350°F.
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 is quick, about 1 minute on each side.
Remove fried Vdolky from the oil carefully and place it onto a rack.
Put a half tablespoon plum butter on the Vdolek surface on the side with a hole. Sprinkle with grated crumbled cheese.