Sprinkle the yeast over the warm water, and let stand for 5 minutes, or until foamy.
In a large bowl, mix together the yeast mixture, milk, sugar, salt, eggs, 70g butter and 1/2 of the flour. Mix for a few minutes at low speed, or stirring with a wooden spoon. Beat in remaining flour a little at a time, until the dough no longer sticks to the bowl. Knead for about 5 minutes, or until smooth and elastic. Place the dough into a greased bowl, and cover. Set in a warm place to rise until double. Dough is ready if you touch it, and the indention remains.
Turn the dough out onto a floured surface, and gently roll out to a 1.5cm thickness. Cut with a floured round doughnut cutter. Let doughnuts sit out to rise again until doubled. Cover loosely with a cloth.
Melt 75g butter in a saucepan over medium heat. Stir in icing sugar and vanilla until smooth. Remove from heat, and stir in hot water 1 tablespoon at a time until the icing is somewhat thin, but not watery. Set aside.
Heat oil in a deep-fryer or large heavy pan to 175 degrees C. Slide doughnuts into the hot oil, working in batches to avoid overcrowding the pan. Turn doughnuts over as they rise to the surface. Fry doughnuts on each side until golden brown. Remove from hot oil and drain on a wire rack.
Dip doughnuts into the glaze while still hot, and set onto wire racks to drain off excess. Keep a baking tray under racks for easier clean up.