Step 1

Melt the butter and warm the milk. Cut 1 stick unsalted butter into large pieces and place in a small microwave-safe bowl. Microwave on high for 1 minute until mostly melted (a few lumps are fine) and set aside. Microwave 1 cup whole milk in a microwave-safe mug or measuring cup for 1 minute until warm but not hot (about 100ºF).

Step 2

Proof the yeast in the warm milk. Transfer the warm milk to a stand mixer (or a large bowl) and whisk in 1/4 cup granulated sugar. Sprinkle 1 packet (2 1/4 teaspoons) active yeast over the milk and set aside for 5 minutes. The yeast should get bubbly and look like foam. (If it doesn't, get new yeast and start over.)

Step 3

Whisk the butter and eggs into the milk mixture. Pour the cooled butter into the milk mixture and add 2 large eggs. Whisk to combine and break up the eggs.

Step 4

Add the flour and salt and knead the dough into a smooth, tacky dough. Add 4 cups all-purpose flour and 3/4 teaspoon kosher salt. Using the dough hook attachment, beat the dough on low speed until it comes together, about 3 minutes. Increase the speed to medium-high and continue to beat until the dough is smooth and tacky, about 3 minutes more. (If kneading by hand, stir in the flour and salt until a sticky dough forms, then knead on a floured work surface until the dough is smooth and tacky, about 5 minutes.)

Step 5

Form the dough into a ball, cover, and refrigerate for 8 to 12 hours. Remove the dough from the bowl and shape it into a tight, smooth ball. Sprinkle the dough with a little additional flour (about 1 tablespoon) and return to the bowl. Cover with a clean kitchen towel or plastic wrap and refrigerate until doubled in bulk, 8 to 12 hours. (Alternatively, you can just let the dough rose at room temperature for 1 1/2 to 2 hours before frying — the finished donuts with be denser but still delicious.)

Step 6

Roll out and cut the dough. Coat a work surface with about 1 tablespoon all-purpose flour, then transfer the dough onto it. Pat the dough down into a thick round, then use a rolling pin to roll it out about 1/3-inch thick. Using a donut cutter or a 3-inch round cutter, cut out as many rounds as possible. Use a 1-inch round cutter to cut out the holes if needed. Do not reroll the scraps. Transfer the donuts and donut holes to 2 lightly floured baking sheets. Cover with a clean kitchen towel and let rise until puffy, at least 30 minutes. Meanwhile, make the glaze.

Step 7

Make the glaze and set up a cooling station. Melt 4 tablespoons unsalted butter in a small saucepan over low heat. Add 2 cups powdered sugar, 1/4 cup whole milk, and 1 teaspoon vanilla extract, and whisk until smooth. Remove from the heat and cover to keep warm. Set up a cooling station: line a baking sheet with paper towels and place a wire rack over the paper towels. This will help wick oil away from the finished doughnuts while keeping them tender crisp.

Step 8

Heat the oil for frying. Wait until the doughnuts have been rising for about 30 minutes and are starting to look puffy before heating the oil. Pour 2 quarts of canola oil into a large heavy-bottomed pot or Dutch oven and attach a deep-fry thermometer. Heat over medium-high heat until the oil reaches 355 to 365ºF. Reduce the heat slightly to maintain this heat.

Step 9

Fry the doughnuts. Fry in batches of 4 doughnuts. You can transfer the doughnuts using a pair of chopsticks (my favorite tool for flipping and removing the doughnuts too) or with a flat spatula. Fry until puffed and golden brown, about 1 1/2 minutes per side. Use tongs or chopsticks to remove the doughnuts from the hot oil, hold them over the pot for a minute to drain off excess oil, then place on the rack. Repeat with the remaining doughnuts, adjusting the heat as needed to maintain an oil temperature between 355 to 365ºF.

Step 10

Glaze the doughnuts. While the donuts are warm, carefully dunk them in the glaze. You can coat just one side or both. If the glaze has cooled and thickened, place back over low heat and stir until loose again. Set the glazed doughnuts back on the cooling rack and let the glaze set for 3 minutes before serving.

Step 11

Fry and glaze the doughnuts holes. Fry the doughnut holes until browned, about 1 minute. Transfer to the rack and cool for 1 minute before glazing.