In a medium bowl, dissolve the yeast in the warm water.
Add the milk, margarine or butter, egg (yolk & whites, please/ no shell), sugar, and salt, and blend with an electric mixer until smooth.
Add about half the flour and mix for 30 seconds.
Add the remaining flour and knead the dough with flour-dusted hands until smooth.
Cover the bowl of dough and leave it in a comfy, warm place until the dough doubles in size, about 1 hour. You can tell that the dough has risen enough when you poke it with your finger and the indentation stays.
Roll out the dough on a heavily floured surface until it's about 1/2 inch thick.
When you've punched out all the dough (you should have about a dozen unholed donuts), it's time for the holes. Find the cap to a bottle of lemon juice or Worcestershire sauce, or any other small cap with a diameter of about 1 1/4 inches. Use this to punch out holes in the center of each of your donuts
Place the donuts on plates or cookie sheets, cover, and let stand in the same warm, comfy place until they nearly double in size. This will take around 30 to 45 minutes.
Heat the vegetable oil in a large frying pan over medium heat. Bring the oil to about 350° F. it is easily tested with tiny portions of scrap dough left over from punching out the doughnuts. The dough should be bubbling rapidly, immediately from when you drop it in.
Fry each donut for about 30 seconds per side, or until light golden brown. Cool 5 minutes on paper towels, unless you want to powder them, and if so, then you want to cover them in powder after 30 seconds of cooling. You are done if you made powdered donuts at this point.
For either the plain or the chocolate glaze, combine the margarine or butter with the powdered sugar in a medium bowl and blend with an electric mixer.
Add the vanilla and hot water. Mix until smooth.
If you're making the chocolate glaze, melt the chocolate chips in a microwave-safe bowl in the microwave for 30 to 40 seconds. Stir, then microwave another 30 seconds and stir again until completely melted. Add to the plain glaze mixture. Blend until smooth.
When the donuts have cooled, dip each top surface into the glaze and then set on a plate to cool for about 15 minutes, until the glaze firms up.Makes 12 Source: