Mix yeast in 1 Celsius of the flour, add sugar. Combine water, milk, melted butter, and egg in the mixer, add the flour/yeast/sugar mixture blending well (about 3 minutes). Add vanilla, salt and remaining flour. The dough should be quite soft, but not sticky.
Cover the dough and let rise until double. (The author generally uses the moist oven method for faster results: Preheat the oven to 200F, place a loaf pan filled halfway with boiling water in the oven. SHUT OVEN OFF.
Place dough in warm, moist oven to rise. This method generally reduces rising time by half.) Dough has risen enough when an indentation remains when you poke it.
On a heavily floured surface, roll dough out 1/2" thick and cut out the donuts using a donut cutter dipped in flour.
Let the donuts and holes rise/set for 15 minutes. In the meantime, heat up the oil for frying, 3-4" deep. The oil needs to be about 360F in order to prevent it from saturating the dough. A drop of dough should sizzle. Fry for approximately 60 seconds each side, turning only ONCE. Remove from hot oil, place on paper towels, cool for 5-7 minutes before glazing.
To make glaze: Combine butter, confectioners sugar, vanilla, honey and hot water in a small bowl, blending until smooth. Dunk the donuts into the glaze.