In the bowl of a stand mixer, mix the yeast and 1/2 cup of the lukewarm milk. Let stand for 5 minutes, then stir in 1 tablespoon of the sugar and 1/2 cup of the flour. Cover with plastic wrap and let it stand at room temperature for 20 to 30 minutes, until bubbles form.

Reheat the remaining 1/2 cup milk to lukewarm. Stir in the milk, buttermilk, nutmeg, zest, and remaining 1/2 cup plus 1 teaspoon sugar. Place the remaining flour and the salt on top of the sponge. Put the bowl on the stand mixer, and using the paddle attachment, mix on low speed for 2 minutes. Scrape down the sides of the bowl and change to the dough hook. Increase the speed to medium and knead the dough for 5 minutes. The dough should be tacky.

Turn the speed to medium-low and add the butter, one tablespoon at a time. Scrape down the sides of the bowl. Mix for 5 minutes, or until the dough is smooth. Scrape down the sides of the bowl again and cover the bowl with plastic wrap. Let the dough rise at room temperature until doubled – 1 1/2 to 2 hours.

When the dough has doubled, turn it out onto a slightly floured surface. Roll into a 10-inch square and wrap with plastic wrap. Place it in the freezer for 20 minutes.

Take out of the freezer, and using doughnut cutter, cut out the doughnuts. Dip the cutter in flour each time to make cutting easier. Place the doughnuts on a sheet pan covered with parchment paper. Cover with lightly sprayed plastic wrap and let rise for 30 minutes.

In a deep heavy skillet or deep saucepan fitted with a deep-fry thermometer (or a deep fryer) heat oil to 350F. Cut the parchment that the doughnuts are sitting on into squares, each square holding a doughnut or a doughnut hole. Carefully slide 4 doughnuts off their parchment into the oil. After 30 seconds, turn the doughnuts over, cook for 1 minute, then flip it back over a cook for an additional 30 seconds. Remove from oil and let them drain on paper towels. Once they are cool enough to handle, dip them in the prepared glaze (recipe below). Repeat with the remaining doughnuts and doughnut holes.

Combine milk and vanilla in a medium saucepan and heat over low heat until warm.

Sift confectioners’ sugar into milk mixture. Whisk slowly, until well combined.

Remove the glaze from the heat and set over a bowl of warm water.

Dip doughnuts into the glaze, 1 at a time, and set on a draining rack placed in a half sheet pan for 5 minutes before serving.