Dissolve yeast in 1 cup of warm water. Let sit about 5 minutes until the yeast foams a bit (so you know it's active).
Beat in 2 cups of the flour, sugar, egg, butter, and salt. Add the remaining 1 1/4 cup flour 1/4 cup at a time until the dough starts pulling away from the sides of the bowl. Cover bowl with plastic wrap or a clean kitchen towel and let sit in a warm place until the dough doubles in bulk, about 2 hours. Alternatively, you can chill the dough overnight.
Punch down the dough and turn it onto a floured surface. Roll dough to about 1/2-inch thick and use a large round biscuit cutter (a drinking glass works too) to cut circles and a smaller round cutter to cut out the holes. Obviously, if you are lucky enough to be in possession of a doughnut cutter, go ahead and use that!
Cover doughnuts (and holes) and let rise until puffy looking, about 2 hours. Alternatively, you can put the cut doughnuts on floured baking sheets, cover, and chill the cut dough overnight.
Heat an inch or two of oil or lard in a large heavy pot to 350 - 375*F. Set a cooling rack over a baking sheet and put them near the pot. Put about a cup of sugar in a medium bowl and have that handy as well.
Add 3 or 4 doughnuts (or 8 to 10 doughnut holes) to the oil. They should sizzle immediately as you add them to the fat. Cook doughnuts until light brown on one side. Use tongs or a slotted spoon to flip the doughnuts, cook until the second side is light brown. Transfer cooked doughnuts to cooling rack and let cool/drain for a few minutes. Dip doughnuts into sugar to coat. Repeat with remaining doughnuts. Serve warm if at all possible.
Makes about 18 homemade doughnuts and doughnut holes.