Mix together the milk and yeast in a small bowl and set aside for a couple minutes.
Mix together the flour, mashed potatoes, butter, limoncello if using, vanilla, salt, egg, lemon zest, the milk and yeast mixture and 1/4 cup of the sugar in a large bowl until it forms into a dough (it will be sticky at this point).
Dump it onto a well-floured surface and knead until the dough comes together and is nice and smooth and no longer sticky (this might cause you to use a bit more flour than you might expect, so keep some on hand).
Once your dough is ready, line a large baking sheet with a lint-free kitchen towel and sprinkle the towel lightly with flour.
Cut little pieces (the size you cut depends on how big you want the doughnuts to be; I make them small, so I cut pieces as big as a golf ball) of the dough and roll each one out into a 6-inch rope, then pinch the ends together to form a circle. Place on the baking sheet, leaving a few inches of space between. Continue forming the rest and once you're all done, cover them with another kitchen towel and allow them to rise in a warm spot until they have doubled in volume, about an hour.
Put the remaining 2 cups sugar in a baking dish. Pour enough oil into a large pot to come up the sides a few inches and heat until a deep-frying thermometer registers 350 degrees F. Drop about 4 doughnuts at a time in the oil and cook until golden brown, turning once, about a minute per side. Drain on a paper-towel lined plate, then immediately roll them in the sugar to coat. Repeat with the remaining doughnuts.