Grease a medium bowl thoroughly and set aside.

Warm the milk to 115 degrees (F) and place in your large mixer bowl if you're using an electric mixer, or a large mixing bowl if you're mixing by hand.

Add the sugar and yeast. When the mixture bubbles (be patient--this can take up to 15 minutes), add the butter, egg, salt, and flavoring extract if using. Add one third of the flour. Mix by hand or with your mixer until the flour is incorporated. Similarly, add the next 1/3 and then the last one 3rd of the flour. Mix until the dough is smooth. If dough is too sticky add additional flour, 1 tablespoon at a time. Dough should be tacky when touched.

Tightly covered the dough with a damp cloth and then plastic wrap and allow the dough to rise and puff up--an hour or two, depending on the warmth of the room.

Coat your hands with flour. Lightly squish the dough into a ball and place it in the greased bowl you have prepared. Move the dough ball around to grease its outside. Again, cover the bowl tightly with a damp cloth and then a layer of plastic wrap.

Allow the dough to rise again for 1 to 2 hours in a warm place.

You can refrigerate the dough for as much as 48 hours. Remove it from the refrigerator an hour before forming.

When you're ready to form the beignets, coat your hands with flour and lightly knead the pumpkin pie spice into the dough, not uniformly. You want to end up with swirls of spice. Don't overwork the dough.

On a lightly floured pastry cloth, roll the dough to about 1/4-inch thick. If the dough has toughened too much to roll it thinly, cover it with a damp cloth and leave it alone for about 10 minutes to allow the dough to loosen, and then roll it to 1/4-inch thick. Cut in strips about 2 inches wide, and cut the strips into approximate squares. Some of them will end up as triangles, and that's fine. Cover the uncooked beignets with the damp cloth and allow them to rest while you heat the oil.

Pour 1 to 2 inches of oil into a deep, cast-iron skillet set on your stovetop. For safety's sake, put a lid that fits on the skillet nearby in case the oil overheats and flames up--this should not happen, but if it does, extinguish the fire with the lid. Set the burner's heat to medium and slowly heat the oil to 350 degrees (F). Again, be patient.

Or you can use a deep fryer, following the manufacturer's suggestions.

Using a spatula, gently slip the pieces of dough into the hot oil, being careful not to splash. Fry in small batches. Do not overcrowd, and keep an eye on the temperature of the oil, adjusting the heat as necessary. Fry for about 1-1/2 minutes until golden brown, turn with tongs, and fry an additional 1-1/2 mintues.

Lift with tongs and drain on paper towels.

While the beignets are warm, press powdered sugar through a sieve onto them or shake them with powdered sugar in a clean paper bag. Some of the sugar will soak into the warm beignets. If you like, you can add more as it cools. If possible, serve the beignets while they're warm.