Place the warm milk in a bowl and whisk in the yeast. Add the warm water and let it stand five minutes or until the yeast starts to bubble and rise.
In the bowl of a stand mixer, add the flour, sugar, and salt. Whisk using the paddle attachment. Then add the yeast-milk mixture, vanilla extract and bean scrapings, if using, and applesauce or aquafaba. Whisk until the dough forms.
Add the olive oil in three batches, letting it incorporate into the dough each time before adding more.
After the oil is thoroughly incorporated, swap out the paddle attachment for a dough hook.
Knead the dough on medium speed for 10 minutes or until the dough forms a ball. It may still be a bit sticky and some dough may stick to the bottom.
Scrape out all of the dough into a large greased bowl. Let the dough stand, covered, for about 30 minutes, then turn it over once so the top is smooth. Lightly oil the top, then cover the dough with an oiled plastic wrap so the wrap is directly touching the surface of the dough. Refrigerate for four to 16 hours (overnight is fine). The long proofing time helps the flavor develop.
Remove the dough to a floured surface. Shape it to some evenness with your hands. Then, using a floured rolling pin, roll it into a rectangle about 9 by 10 inches. Cover with a light towel. Let the dough relax for 10 minutes.
Meanwhile, cut out 18 five-inch square pieces of parchment paper a couple of inches wider than the width of your donut cutter. Place them on a baking sheet one beside the other, and spray lightly with cooking spray. This makes it much easier to transfer the risen doughnuts to the hot oil without deflating them.
Then, using a doughnut cutter, cut out doughnuts and doughnut holes. Roll up the scraps and cut out more. Place each doughnut on a square of parchment paper, and put the doughnut holes on another sheet of parchment paper. Each square should accommodate three to four doughnut holes. Leave some room between the holes and the doughnuts because they will rise.
Cover the doughnuts with a light towel and let them rise in a warm place. In warm temperatures, they should double in size in no more than an hour. It might take up to three hours in a cooler place.
Heat oil for frying in a wok or a large dutch oven. If you have a frying thermometer or a candy thermometer, use it to ensure that your oil reaches the optimal temperature for frying: 325 degrees.
While the oil is heating, make the glaze. Place all ingredients in a bowl large enough to dip the doughnuts, then whisk until very smooth. Cover and set aside while you fry the doughnuts.
Fry the doughnuts and the holes, without crowding them. Do this by picking up the parchment paper and putting the whole thing in the frying pan, to avoid deflating the donut. The paper will float free almost immediately, at which point just remove it with a pair of tongs. Fry them for a minute on each side, flipping over twice, for four minutes total.
Place the fried donuts on a rack with a cookie sheet under them. Let them cool for a couple of minutes, then dip each doughnut and doughnut hole in the glaze, turning over once. Place back on the rack. Eat the doughnuts as soon as the glaze is set.