In a large deep pot or fryer heat the oil to 350 degrees F.
In a small mixing bowl, whisk the yeast, 1 teaspoon sugar, and 1/4 cup of the warm water. Set the mixture aside. In an electric mixer, and the eggs and whip until the eggs are thick and pale yellow in color. Change the mixer attachment to a dough hook. With the machine running slowly add the yeast mixture, melted butter, milk, and half and half. Add the salt. Add the flour, 1 cup at a time. Mix until the mixture forms a soft ball, leaves the sides of the bowl, and climbs up the dough hook. Remove the dough and turn into a lightly oiled bowl. Cover the bowl with plastic wrap and set the bowl in a warm, draft free space. Let the dough rise until double in size, about 1 1/2 hours.
Turn the dough out onto a floured surface and dust the surface of the dough with flour. With a rolling pin, roll out the dough, about 1/4-inch thick, in the shape of a rectangle. Cover the dough with a greased piece of plastic wrap (this will prevent the dough from sticking to the wrap) let the dough rise until double in size, about 1 hour. Using a sharp French knife, cut the dough into 1 inch squares. Fry a couple of the squares at a time, until golden brown, about 3 to 4 minutes, stirring constantly for overall browning. Remove the doughnuts from the oil and drain on a paper-lined plate. Roll the doughnuts in sugar and serve warm.