Step 1In a small bowl, combine one-fourth cup warm water and 1 teaspoon of the sugar. Sprinkle the yeast over the mixture. Let sit 5 to 6 minutes; if the yeast is not absorbed, stir lightly until the mixture is creamy.
Step 2Combine the remaining one-half cup sugar, salt and 3 cups of the flour in a large bowl with a whisk, or in the bowl of a stand mixer with the whisk attachment.
Step 3Combine the milk and butter in a small saucepan and heat over low heat just until the butter is melted.
Step 4If you are using a standing mixer, swap out the whisk attachment for the dough hook. In a slow, steady stream, mix the milk mixture into the sugar-salt-flour mixture. If mixing by hand, stir with a fork or wooden spoon. Add the egg, the yeast mixture and the remaining 1 cup of flour. Mix until a soft dough forms (you may need to scrape down the sides of the bowl with a rubber spatula). Cover tightly with plastic wrap and set aside in a warm spot until the dough doubles, 1 to 2 hours.
Step 5Heat about 3 inches of oil in a deep fryer or Dutch oven until the temperature reaches about 370 degrees.
Step 6Divide the dough into thirds. Working in batches on a floured work surface, knead each piece briefly. Using a rolling pin, roll the dough out into a rectangle less than one-fourth-inch thick. Cut the dough into roughly 3-inch squares and gently drop them, 2 or 3 at a time, into the hot oil. Cook 1 to 2 minutes, until golden brown, then flip them over carefully with a slotted spoon. Cook 1 to 1 1/2 minutes longer, until the beignets are puffed and evenly browned. Using a slotted spoon, transfer the beignets to paper towels to drain off excess oil. Dust thickly with powdered sugar and serve hot or warm.