Bring the milk to a simmer in a 1-quart heavy saucepan, then remove from the heat and stir in the sugar and salt. Pour into a large bowl and let cool slightly.
While the milk is cooling, dissolve the yeast in 2 tablespoons warm water in a small bowl, stirring. Let stand for 5 minutes.
Stir the yeast mixture into the milk mixture along with the oil, lemon zest and eggs. Stir in 2 1/2 cups of the flour with a wooden spoon to make a very soft dough. Spread 1 cup of the flour on a work surface and put the dough on top, scraping it from the bowl with a rubber spatula. Knead the dough for about 5 minutes, incorporating all of the flour from the work surface and adding just enough additional flour (if necessary); the dough should be slightly sticky. Transfer the dough to an oiled large bowl and sprinkle lightly with additional flour. Cover the bowl with a clean kitchen towel and let the dough rise in a warm draft-free place until doubled in bulk, 1 to 1 1/2 hours.
Turn the dough out onto a floured surface and roll out with a floured rolling pin until 1 inch thick. Cut out rounds with a 2-inch cutter. To make ring shapes, cut out doughnut holes with a 1-inch cutter. Place on a baking sheet, cover the doughnuts with another kitchen towel and let rise in a warm place for 30 minutes.
While the doughnuts rise, heat about 10 cups vegetable oil in a deep 4-quart pot until it registers 365 degrees F on a deep-fry thermometer. Fry the doughnuts 2 at a time, turning once or twice, until puffed and golden brown, 2 to 3 minutes per batch. Transfer to paper towels to drain.