1In a large bowl, stir together the water, a spoonful of the sugar and the yeast and let it sit for 10 minutes. If it's not foamy, toss it and buy new yeast.
2In a small bowl, whisk together the milk, eggs and vanilla. Add to the yeast mixture along with 3 cups of the flour, the remaining sugar, the butter and salt and stir to combine. Gradually add another 4 cups of flour, stirring (or letting your dough hook do the work) until you have a sticky dough. If it's not too sticky to knead, knead it on a floured surface or with the dough hook; if it's still pretty sticky, add another 1/2 cup flour. Knead until it's smooth and elastic, yet still tacky.
3Put the dough into an oiled bowl, cover with a tea towel and let rise for 1 1/2-2 hours.
4When you're ready to cook, heat a couple inches of oil in a deep-fryer or heavy saucepan until hot, but not smoking (about 350F).
5Roll the dough out to about 1/2-inch thick and cut into 1-2 inch squares with a knife. Carefully drop a few at a time into the hot oil, without crowding it (cooking too many at a time will bring the temperature of the oil down) and cook until golden on both sides, carefully flipping with tongs as necessary. Transfer to paper towels to drain and cool, then douse in powdered sugar.
6Makes about 4 dozen beignets.