Step 1

FOR THE DOUGHNUTS: Stir flour, sugar, and yeast together in bowl of stand mixer. Add milk and egg and mix with rubber spatula until all ingredients are moistened. 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.

Step 2

Add salt and mix on medium-low speed until dough is smooth and elastic and clears sides of bowl, 5 to 7 minutes. 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. Transfer dough to lightly greased large bowl, flip dough, and form into ball. Cover bowl with plastic. Let sit at room temperature for 1 hour. Transfer to refrigerator and chill overnight (or up to 48 hours).

Step 3

Adjust oven racks to lowest and middle positions. Place loaf pan on lower rack. Line rimmed baking sheet with parchment paper and grease parchment. Transfer dough to lightly floured counter. Press into 8-inch square of even thickness, expelling as much air as possible. Roll dough into 10 by 13-inch rectangle, about ½ inch thick. Using 3-inch round cutter dipped in flour, cut 12 rounds. Using 1-inch cutter dipped in flour, cut hole out of center of each round. Transfer doughnuts and holes to prepared sheet. (If desired, use 1-inch cutter to cut small rounds from remaining dough. Transfer to sheet with doughnuts.) Bring kettle or small saucepan of water to boil.

Step 4

Pour 1 cup boiling water into loaf pan. Place sheet on upper rack, uncovered. Close oven and allow doughnuts 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.

Step 5

FOR THE GLAZE: Whisk sugar, water, and salt in medium bowl until smooth.

Step 6

About 20 minutes before end of rising time, add oil to large Dutch oven until it measures about 1½ inches deep and heat over medium-low heat to 360 degrees. Set wire rack in second rimmed baking sheet and line with triple layer of paper towels. Using both your hands, gently place 4 risen doughnuts in oil. Cook until golden brown on undersides, 1 to 1½ minutes, adjusting burner as necessary to maintain oil temperature between 350 and 365 degrees. Using spider skimmer, flip doughnuts and cook until second sides are browned, 1 to 1½ minutes. Transfer doughnuts to prepared rack. Return oil to 360 degrees and repeat with remaining doughnuts. For doughnut holes, transfer all to oil and stir gently and constantly until golden brown, about 2 minutes. Transfer to prepared rack to cool. Let doughnuts sit until cool enough to handle, at least 5 minutes.

Step 7

Set clean wire rack in now-empty sheet. Working with 1 doughnut at a time, dip both sides of doughnut in glaze, allowing excess to drip back into bowl. Place on unlined rack. Repeat with doughnut holes. Let doughnuts and holes stand until glaze has become slightly matte and dry to touch, 15 to 30 minutes, before serving.