In a large bowl, stir together the warm water, warm milk, and active dry yeast. Let it sit for about 5 minutes until the mixture becomes foamy. This means your yeast is alive and ready.

Add sugar, salt, egg, shortening, and flour to the yeast mixture. Use a hand or stand mixer with the dough hook and beat on medium speed for about 2 minutes, or until the dough is well combined and pulls away from the sides of the bowl.

Lightly spray a large glass or ceramic dish with nonstick cooking spray. Transfer the dough to the dish, cover it with plastic wrap or a clean kitchen towel, and let it rise in a warm spot until it doubles in size—about 45 minutes to 1 hour.

Once the dough has risen, roll it out on a lightly floured surface to about ½-inch thickness. Use a donut cutter or two round cutters (one large, one small) to cut out your donut shapes. Place them on a parchment-lined baking sheet, cover lightly, and let them rest for 15 minutes.

Heat oil in a deep fryer or large, heavy-bottomed pot to 350°F. Fry donuts in small batches for about 40–60 seconds on each side or until they are golden brown. Use a slotted spoon to remove them and transfer to a paper towel-lined tray to drain.

In a medium bowl, whisk together melted butter, powdered sugar, vanilla extract, and hot water until smooth. The glaze should be thin enough to dip but thick enough to coat.

While the donuts are still warm, dip each one—front and back—into the glaze. Place on a wire rack set over parchment or a baking sheet to catch drips. Let them set for a few minutes before serving.