Stir flour, sugar, and yeast together in bowl of stand mixer. 4 ½ cups all-purpose flour, ½ cup granulated sugar, 1 teaspoon instant yeast
Add milk and egg and mix with rubber spatula until all ingredients are moistened. 1 ½ cups whole milk, 1 large egg
Fit stand mixer with dough hook and mix on medium-low speed until cohesive mass forms, about 2 minutes, scraping down bowl if necessary. Cover bowl with plastic wrap and let stand for 20 minutes. Add salt and mix on medium-low speed until dough is smooth and elastic and clears sides of bowl, 5 to 7 minutes.
1 ½ teaspoons kosher salt
With mixer running, add butter, a few pieces at a time, and continue to mix until butter is fully incorporated and dough is smooth and elastic and clears sides of bowl, 7 to 13 minutes longer, scraping down bowl halfway through mixing.
8 tablespoons unsalted butter
Transfer dough to lightly greased large bowl, flip dough, and form into ball.
Cover bowl with plastic wrap. Let sit at room temperature for 1 hour. You can choose to skip to step 4 after 1 hour of rising, or transfer the dough to the refrigerator overnight (or up to 48 hours).
After the dough has chilled, but before you remove the dough from the fridge, prepare for the next step by adjusting oven racks to lowest and middle positions.
Place loaf pan on lower rack.
Cut out 14 (6x6-inch) square pieces of parchment paper (12 for the donuts, 2 for the donut holes) and place on prepared baking sheets.
Spray squares with nonstick cooking spray and set aside.
Transfer chilled dough to lightly floured counter. Press into an 8-inch square of even thickness, expelling as much air as possible. Roll dough into a 10 by 13-inch rectangle, about ½ inch thick. Using a 3-inch round cookie or biscuit cutter dipped in flour, cut out 12 rounds. Using a 1-inch cookie cutter dipped in flour, cut a hole out of the center of each round. Transfer donuts and holes to the prepared baking sheets, placing one donut on each square. (If desired, use 1-inch cookie cutter to cut small rounds from remaining dough to create additional donut holes. Transfer to baking sheet with donuts.) Meanwhile, boil one cup of water in a saucepan or microwave. Pour the 1 cup boiling water into the loaf pan that's set on the lower rack of your oven. Place baking sheets on the upper rack of your cool oven, uncovered. Close oven door and allow donuts to rise until dough increases in height by 50 percent and springs back very slowly when pressed with your knuckle, 45 minutes to 1 hour.
Meanwhile, prepare the glaze by whisking together the powdered sugar, water, and salt in medium bowl until smooth. Set aside.
3 ¼ cups powdered sugar, ½ cup hot water, pinch of kosher salt
About 20 minutes before the end of the second proof time, add vegetable oil to a large Dutch Oven until it measures about 1½ inches deep and heat over medium-low heat to 360ºF (182ºC).
2 quarts vegetable oil
Set wire rack in on top of a rimmed baking sheet and line with a triple layer of paper towels to absorb the frying oil from the donuts. Use parchment paper squares to gently pick up donuts, one at a time, and add to oil, being careful not to let the parchment get in the oil. Fry until golden brown on undersides, 1 to 1½ minutes, adjusting burner as necessary to maintain oil temperature between 350ºF (177ºC) and 365ºF (185ºF) degrees. Using a spider skimmer, flip donuts and fry until second sides are browned, 1 to 1½ minutes. Transfer donuts to prepared wire rack using spider skimmer. Return oil to 360ºF and repeat with remaining donuts.
For donut holes, transfer all to oil and stir gently and constantly until golden brown, about 2 minutes. Transfer to prepared rack to cool.
Let donuts and donut holes sit until cool enough to handle, at least 5 minutes, before glazing.
Set clean wire rack on top of the baking sheet where the donuts had proofed. Give the glaze a good whisk. Working with 1 donut at a time, dip both sides of the donut in the glaze, allowing excess to drip back into bowl. Place on unlined rack. Repeat with donut holes.
Let donuts and holes stand until glaze has become slightly matte and dry to touch, 15 to 30 minutes, before serving.