Louisiana made the beignet their official state doughnut. They're a treat for both breakfast and dessert--just make sure they're served hot and fresh. Martha made this recipe on Martha Bakes episode 602.
Using a standing mixer fitted with a dough hook, mix together yeast, warm water, and granulated sugar. Let stand until foamy, about 5 minutes. In a separate bowl, whisk together flour, salt, and nutmeg. While mixing yeast mixture on medium speed, add butter. In a smaller bowl, lightly whisk together milk and egg using a fork before adding to the yeast-butter mixture. Mix in 1 1/2 cups flour mixture to combine, then add 1 3/4 cups more flour mixture and mix until thoroughly incorporated. Turn out dough onto a lightly floured surface. Knead in remaining 1/4 cup flour mixture by hand until dough is smooth, about 5 minutes.
Place dough in a lightly oiled bowl, cover loosely with plastic wrap, and let stand in a warm, draft-free spot until doubled in size, about 1 hour. Turn out dough onto a lightly floured surface and punch down. Using a lightly floured rolling pin, roll out dough to a 12-inch square. Using a pizza wheel or sharp knife, cut dough into 3-inch squares. Transfer squares to a floured baking sheet and cover loosely with plastic wrap. Let rise in a warm, draft-free spot for 30 minutes.
Meanwhile, heat oil in a medium pot or deep-fryer until it registers 350 degrees on a deep-fry thermometer. Working in batches, add a few squares to the oil and fry, rolling them around constantly with a slotted spoon or spider, until golden brown all over, 1 to 2 minutes. Transfer beignets to a paper towel-lined baking sheet to drain. Coat with confectioners' sugar, and repeat process with remaining dough and more confectioners' sugar. Serve warm.