Combine 1 cup flour, sugar, undissolved yeast, salt and nutmeg in a large mixer bowl. Heat water, milk, and butter until very warm (120o to 130oF). Gradually add to flour mixture. Beat 2 minutes at medium speed of electric mixer, scraping bowl occasionally. Add egg and 1 cup flour; beat 2 minutes at high speed. Stir in enough remaining flour to make a soft dough. Knead on lightly floured surface until smooth and elastic, about 8 to 10 minutes. Cover; let rest 10 minutes. Roll dough to 1/2-inch thickness; cut with 2-1/2-inch or 3-inch doughnut cutter. Knead together trimmings; re-roll and cut. Place doughnuts on greased or parchment lined baking sheets. Cover; let rise until doubled in size, about 1 hour. Heat at least 2 inches of oil in a deep fryer or deep heavy pan to 375oF. Fry 3 to 5 doughnuts at a time, for 2 minutes or until browned, turning once. Drain on paper towels to remove excess oil. Use tongs to dip doughnuts into Powdered Sugar Glaze while warm. Cool on wire racks. To make Powdered Sugar Glaze: Combine powdered sugar, milk and vanilla in a medium bowl. Stir until smooth.