Mix yeast, water, and sugar in a mixer bowl fitted with a dough hook. Let stand until foamy, about 5 minutes. Add salt, nutmeg, egg, milk, and 1 1/2 cups flour; mix on medium speed until combined. Add butter; mix until incorporated. Add 1 3/4 cups flour, and mix until dough comes together. Turn dough onto a lightly floured surface. Knead in remaining 1/4 cup flour 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. Remove dough from bowl; punch down. Roll out on a lightly floured surface to 1/2-inch thickness. Cut out circles with a 1-inch round cutter; place on a floured baking sheet. 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 candy thermometer. Fry beignets a few at a time, rolling them around constantly with a slotted spoon, until golden brown all over, 1 to 2 minutes. Using the spoon, transfer beignets to a paper-towel-lined plate to drain. Dust with confectioners' sugar, and serve warm. Repeat.
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