In the bowl of a mixer, combine 4 cups flour and salt. Add the yeast. In a small bowl, combine the milk and 2 tablespoons sugar. Add with the remaining milk to the flour. (If using dry yeast, mix the yeast with 2 tablespoons of the milk and 2 tablespoons sugar and let sit until foamy, about 5 to 10 minutes and add to the flour.)

Add the eggs and butter to the flour mixture. Begin to mix the ingredients into a soft, but not sticky, dough. Add a little extra milk, 1 tablespoon at a time. Work in extra flour, 1 tablespoon at a time. Turn the dough out onto a lightly floured surface and knead until smooth and elastic. (You can do this in a mixer fitted with a dough hook.) Transfer the dough to a lightly oiled bowl, cover, and let rise until doubled in size, about 2 hours.

Punch down the risen dough. Turn out onto a lightly floured surface and knead for a few seconds. With a lightly floured rolling pin, gradually roll out the dough to about 1/2-inch thick. When rolling dough, let it rest periodically to relax. Cut out into 3 to 4-inch rounds with a lightly floured biscuit cutter. Re-roll the scraps to make more rounds. Place the doughnuts on lightly floured baking sheets, spacing them apart, and cover lightly with a dry towel. Let rise in a warm spot until doubled in size, about 20 minutes.

Transfer the risen doughnuts to a heavy pot filled 3-inches high with vegetable oil at 350 degrees F. Fry the doughnuts, a few at a time, until golden and puffed, turning frequently, 5 to 7 minutes.