In a medium bowl, stir the yeast and water together. Set aside to allow the yeast to proof.
After 10 minutes, stir in the all-purpose flour, cover with plastic, set aside in a warm place and let it double in size.
In the bowl of an electric mixer fitted with a paddle attachment, cream together the butter and sugar for 8 to 10 minutes. Add the eggs, with the mixer on low speed, one at a time. When the mix is smooth, add the vanilla and salt. Stir to blend. Add the bread flour and mix on low speed until fully incorporated.
Change the electric mixer attachment to a dough hook. Continue mixing the dough until it becomes a ball. Then, stir in the yeast mixture until thoroughly incorporated. Put the dough in a bowl, cover with plastic. Let it rise for 1 hour to 1 and a half hours.
Lightly flour a cool surface and roll the dough out to about 1 1/2-inch thickness. Cut rounds of dough with a cutter and turn them onto a floured baking sheet. Cover them with a dish towel and let them rest, 20 to 30 minutes, before frying.
Heat the oil in a large pot to 365 to 370 degrees F. Drop the beignets in the hot fat and cook until golden brown on both sides, 3 to 4 minutes. (If they seem to be browning too quickly reduce heat slightly.) Remove them from the oil with a slotted spoon. Roll them in the granulated sugar to coat and set aside.