Place 4 tablespoons unsalted butter in a small microwave-safe bowl. Microwave on HIGH until melted. Set aside to cool slightly.

Place 3 ½ cups all-purpose flour, 2 teaspoons baking powder, 1 teaspoon kosher salt, ¾ teaspoon ground nutmeg, and ¼ teaspoon baking soda in a medium bowl and whisk to combine.

Place the melted butter, ½ cup packed light brown sugar, ½ cup granulated sugar, and 1 tablespoon vanilla extract in the bowl of a stand mixer fitted with the paddle attachment. Beat on medium speed until combined, scraping the bowl with a flexible spatula as needed, about 2 minutes.

Reduce the speed to low. Add 2 large eggs and 1 large egg yolk one at a time, stopping and scraping down the bowl as needed. Add ¾ cup buttermilk and mix until just combined.

Add the flour mixture all at once and mix on the lowest speed just until a few streaks of flour remain. Remove the bowl and continue mixing by hand until there are no dry bits of flour. The dough will be quite sticky. Refrigerate for at least 30 minutes and up to overnight.

Line two rimmed baking sheets with parchment paper and dust it with flour. Heavily flour a work surface and your hands with flour. Transfer half of the dough (about 18 ounces) to the work surface, and sprinkle it with flour. Pat the dough into a 3x24-inch rectangle.

Use a floured bench scraper or knife to cut the rectangle crosswise into 1 ½-inch by 3-inch bars. Dip the bench scraper in flour and very gently press it down the center of each bar lengthwise.

Using a brush, gently brush the excess flour off the bars. Use the floured bench scraper or thin metal spatula to gently transfer each bar to one of the baking sheets, flipping it over so that it is indentation-side down. Refrigerate and repeat with shaping and cutting the remaining dough.

Fill a large, heavy-bottomed pot or Dutch oven with about 2 inches vegetable oil. Heat the oil on medium-high heat until 370ºF.

Remove the first tray of bars from the refrigerator. Frying in batches of 6 to 8, use the bench scraper to carefully transfer the donuts into the hot oil one at a time. Fry until golden brown, about 1 ½ minutes per side. Carefully transfer the donuts to the rack with a slotted spoon.

Make the glaze: Place 2 ½ cups powdered sugar, ¼ cup water, and 1 tablespoon vanilla extract in a large bowl and whisk until shiny and smooth.

Dip a donut indentation-side down halfway into the glaze. Lift it up and allow the excess glaze to drip back into the bowl. Return to the rack glazed-side up and repeat with the remaining donuts. Let sit until the glaze is set before serving.