Sprinkle the yeast over the warm water, and let stand for 5 minutes.
In a large bowl, mix together the yeast mixture, milk, sugar, salt, egg, butter, and 1 cup of the flour. Mix for a few minutes at low speed, or stir with a wooden spoon. Add remaining flour and mix until the dough no longer sticks to the bowl.
Knead until smooth and elastic. Place the dough into a greased bowl, and cover. Set in a warm place to rise until double (1 hour).
Turn the dough out onto a floured surface, and gently roll out to 1/2 inch thickness. Cut with a donut or biscuit cutter. Let doughnuts sit out to rise again until double. Cover loosely with a cloth or piece of plastic wrap.
Heat oil in a deep-fryer or large heavy skillet to 350 degrees. Place a few donuts into the hot oil.Turn doughnuts over as they rise to the surface. Fry doughnuts on each side until golden brown. Remove from hot oil, to drain on a wire rack with paper towels underneath to catch excess oil.
Mix glaze ingredients and dip donuts into the glaze while warm, and set back on the wire racks to drain off excess.