In the bowl of a stand mixer, or in a large bowl if making by hand, combine the yeast, warm water, and about 1 teaspoon of the sugar. Let it sit for 5 minutes, or until the yeast is foaming.

Once the yeast is foaming, add the remaining sugar, warm milk, egg, butter, and salt. Add 1 cup of the flour and start mixing. Continue to add flour until the dough starts to smooth out. The dough will still be sticky and sticking to the sides of the bowl.

Switch to the dough hook and continue to knead for 3-5 minutes. Continue to add flour as the dough kneads, 1 tablespoon at a time, until the dough just starts to pull away from the sides of the bowl. (You doń’t want to add too much flour - the dough will still be somewhat sticky.)

Scrape down the sides of the bowl, then cover the bowl with a towel or plastic wrap that is sprayed with nonstick cooking spray. Let the dough sit until doubled in size, about 1 hour.

Once doubled, generously flour a counter or work surface. Turn the dough out onto the surface and flour the top of the dough. Roll the dough out to about 1/2-inch thickness. Use a donut cutter (or two round cookie cutters) to cut out the donut shapes. Transfer to another area of the counter that has been floured, making sure to separate the donut from the donut hole. Once the donuts are cut, cover with a towel and let them rise again for about 30-45 minutes.

While the donuts are raising, add a few inches of oil in a heavy pot or deep cast iron skillet. Heat to about 350ºF.

Also make the glaze by combining the powdered sugar, milk, and vanilla extract. (The glaze should be fairly thin.)

When the donuts have risen, carefully add 3-4 to the hot oil. Let them cook until browned, then flip and cook the other side. This should take about 1 minute per side.

Once browned and cooked through, transfer to a cooling rack over a baking sheet. Once cool enough to handle, dip in the glaze, then return to the cooling rack for the donut to cool and for the glaze to set up.