In a large mixing bowl combine 1 3/4 cups flour with yeast. In a saucepan heat milk, sugar, butter and salt until butter melts. Stir milk mixture into flour along with eggs, beating with a heavy-duty electric mixer on low for 30 seconds, just to combine. Increase speed to high and beat 3 minutes. Add enough of remaining flour, a little at a time, stirring with a spoon to make a firm but pliable dough. Knead dough on lightly-floured work surface 4 to 5 minutes and transfer to an oiled bowl. Cover and let dough rise until it doubles in size, about 1 hour.
Heat oil in a deep-fryer to 360 degrees. Fry doughnuts, in batches, turning several times, until golden all over. Using a slotted spoon remove doughnuts to a rack to drain. If using Honey Glaze, immediately glaze hot doughnuts; if using Chocolate Glaze, let cool first. Or let doughnuts cool and dredge in confectioners' sugar or cinnamon-sugar.
This recipe yields 1 1/2 dozen doughnuts.