Make the dough:

1. Add flour, warm milk, melted butter, sugar, egg, instant yeast, and salt to the bowl of a stand mixer. Attach the dough hook attachment onto the stand mixer, and turn the mixer on to medium-low speed. Mix for around 5-10 minutes for the dough to form. When ready, the dough should be smooth, elastic, and a little bit sticky and tacky to the touch. 
2. Transfer the dough to a large greased bowl, cover with plastic wrap, then place in a warm area to rise for about 1-2 hours, or until the dough is doubled in size.

Shape the dough:

1. Meanwhile, prepare 2 large baking sheets with 12 small parchment squares about 4x4-inches in size). You can buy these or cut them out of 2 large sheets of parchment paper. Also, prepare your oven for proofing by adding a rimmed baking sheet or a large cake pan to the bottom rack, and filling it with boiling water. This humid and warm environment is crucial for the final rise.
2. Turn the proofed dough out onto a lightly floured work surface and roll out to be about ½-inch thick. Using a donut cutter (or a large 3-inch cookie cutter coupled with a small 1-inch cookie cutter) cut out your donuts, then transfer each donut onto a parchment square.
3. Transfer the baking sheets into your humid oven, and allow it to rise for about 20 minutes. They should look risen but not doubled in size.

Make the glaze:

1. In a medium mixing bowl, add the confectioners’ sugar, melted butter, milk, hot water, and vanilla. Whisk until smooth and set aside for later.

Fry the donuts:

1. When the donuts are ready, add oil to a large wide pot, such as a Dutch oven, filling it about halfway. Heat the oil over medium-high heat until it reaches between 350°F as read on a cooking thermometer, about 3-4 minutes.
2. Pick up the donuts with the parchment paper attached, and carefully place them into hot oil. Use rubber tongs to remove the parchment papers from the oil. Fry the donuts for about 60-90 seconds per side, or until golden. It is best to fry in batches – I fried mine in 3 batches, as to not drop the oil temperature too much.
3. Place the fried donuts onto paper towel-lined baking sheet to cool off.

Add the glaze:

1. While the donuts are still warm, dunk each donut into the glaze, then place onto a wire rack for the excess to drip off. Allow the glaze to fully harden before enjoying.