In a medium-saucepan, heat the milk to the scalding point but do not boil. Add sugar, salt, nutmeg and vanilla. Cool to lukewarm.
Sprinkle the yeast into warm water and stir until the yeast is dissolved.
Add oil, egg and dissolved yeast to the milk mixture and blend with a spoon. Transfer mixture to a large bowl and add the flour gradually, beating well until mixed. The dough will be very soft. Cover the bowl with a towel and set the bowl in a warm, draft-free place to rise until it doubles in size (about 1 hour).
Lightly flour a rolling board or work surface and gently knead dough. You don’t want to activate gluten strands too much as dough will become tough. Roll dough to about a 1/4-inch thickness in a rectangle of about 12 x 18 inches. Cut dough into 2″ x 3″ rectangles (roughly 36). Don’t worry if they’re not perfectly the same size. Cover with the towel and let rise for about 45 minutes.
Add between 1/3 and 1/2 pound of powdered sugar to a plain brown bag.
Add frying oil to a heavy-bottomed pan. Add a candy or deep-frying thermometer to manage heat. Bring oil to 375 degrees, then carefully add beignet dough pieces to pan. Do not crowd.
Using a slotted spoon, turn beignets after 30 – 45 seconds so that they brown on all sides. Fry until golden-brown then remove and drain on paper towels.
When beignets are drained, toss into paper bag and shake well so that beignets are thoroughly covered with powdered sugar.