Mix farmer's cheese, flour, eggs, sugar, vanilla extract, and salt together in a bowl until a sticky dough forms.
Roll dough into 1-inch balls with wet hands to prevent sticking.
Heat oil in a deep-fryer or heavy saucepan to 375 degrees F (190 degrees C). The oil should be hot but not smoking. It is ready for frying if you see little air bubbles around a wooden spoon placed in the oil.
Fry a batch of the dough balls in the hot oil until golden, about 2 minutes. Place doughnuts on paper towels to soak up excess oil. Repeat with remaining dough. Sprinkle with confectioners' sugar.