These yeast doughnuts are a fall favorite.
Combine milk, 2 teaspoons sugar, and the yeast in a small bowl, and let stand until foamy, about 10 minutes. Whisk egg, water, and vanilla in a bowl.
Combine flour, cinnamon, nutmeg, salt, and remaining sugar in the bowl of a standing mixer fitted with the paddle attachment. With mixer on low speed, add yeast mixture, egg mixture, and butter, beating until just combined. Switch to the dough hook, and beat on medium speed until dough forms a ball, 18 to 20 minutes. Place dough in a large buttered bowl, and cover tightly with plastic wrap. Refrigerate overnight.
Line a baking sheet with parchment paper, and coat parchment with cooking spray. On a lightly floured work surface, roll dough to 1/4 inch thick. Cut out rounds with a 4-inch cutter, then cut out centers with a 1 1/2-inch cutter. Space rings and holes 1 inch apart on prepared sheet. Coat another piece of parchment with cooking spray, and cover doughnuts. Let rise in a warm place for 1 hour.
For the topping: Combine sugar, cinnamon, and salt in a small bowl.
Heat oil in a large pot until a deep-fry thermometer registers 375. Meanwhile, line a baking sheet with paper towels. Working in batches so as not to crowd pot, lower doughnuts (both rings and holes) into hot oil. Fry, turning them as they cook, until golden brown, 1 to 2 minutes total. Drain on paper towels for 2 minutes, then roll warm doughnuts in topping to coat. Doughnuts are best the day they are made, but they can be stored in airtight containers overnight.
This recipe can be doubled easily. You can knead the dough by hand instead of using a standing mixer if desired.