Add the water, yeast and sugar to the bowl of an electric mixer with the dough hook attached. Lightly whisk to combine. Allow the yeast to bloom for 5 minutes, until bubbly. Add the butter in pieces, the egg, milk, and half the flour and mix on low until combined. Add in the remaining flour and salt and mix on low until combined. Increase the mixer to medium and mix for 5 minutes until soft dough is formed. Transfer the dough to a lightly floured surface and knead gently into a ball. Grease a medium bowl with butter, add the dough, turning around to coat. Cover with a kitchen towel or plastic wrap and set aside in a warm place until doubled in size, about 1 1/2 to 2 hours.
While the dough is proofing, make the Anglaise, and set aside.
After the dough has proofed, punch down the dough, cut it in half, and knead gently to form two balls. Leave one ball covered with a clean towel in the bowl, and place the other half on a lightly floured work surface. Use a rolling pin, roll out the dough to a 1/8-inch thickness. Cut the dough using a sharp knife into 2 1/2-inch squares. Repeat with the remaining ball of dough. Allow the beignets to rest at least 10 minutes before frying.
Fill a large pot 1/2 full with vegetable oil and preheat the oil to 350 degrees F.
Fry the beignets in batches of 6 to 8 at a time for 1 minute on the first side until golden and puffed, and turn over to fry on the second side for 1 minute longer. Remove from the hot oil and drain on a paper-lined tray. Dust generously with confectioners' sugar sifted through a fine mesh sieve. Continue with the remaining pieces of dough. Dust the beignets a final time with more confectioners' sugar. Serve the Cafe Au Lait Creme Anglaise alongside for dunking.
Yield: about 6 dozen