In the bowl of a stand mixer fitted with a dough hook, combine flour, sugar, yeast, and salt. Mix briefly to evenly distribute.

Add milk and eggs. Mix on low speed until a rough, shaggy dough forms, scraping down the bowl as needed.

Increase speed to medium-low and knead for 1–2 minutes, until the dough begins to smooth and forms a ball.

With the mixer running, add the softened butter one piece at a time (I generally do about 1 tablespoon), allowing each addition to fully incorporate before adding the next.

Continue kneading for 5–7 minutes, until the dough is smooth, glossy, and pulls away from the sides of the bowl.

Transfer dough to a lightly greased bowl. Cover and let rise in a warm place until doubled in size, about 1–2 hours.

Punch down dough and turn out onto a lightly floured surface. Lightly flour the top of the dough and rolling pin.

Roll dough to ½-inch thickness. Cut donuts using a 3.5-inch round cutter and a 1-inch cutter for the centers. Gather the dough scraps into a loose ball, cover with plastic wrap, and let rest for 10–15 minutes. Then roll out again and cut additional donuts.

Place donuts and donut holes onto baking sheets lined with parchment paper, spacing them at least 2 inches apart. Cover with plastic wrap and refrigerate overnight, 8–12 hours.

Remove donuts from the refrigerator and let rest at room temperature for 20–30 minutes, until slightly puffy but still cool.

Heat 2-3 inches of vegetable oil in a heavy-bottomed pot to 350°F. Set a wire rack inside a large baking pan.

Fry donuts in batches for 2–3 minutes per side, until deep golden brown. The donut holes will cook much faster. Use a slotted spoon to transfer them to the wire rack to cool.

Make the glaze. In a medium bowl, whisk together powdered sugar, unsweetened cocoa powder, salt, and melted butter. Add 3 tablespoons chocolate or regular milk. Continue adding milk, 1/2 tablespoon at a time, until the glaze is thick but pourable. It should ribbon off the whisk and slowly settle back into the bowl.

Dip the tops of completely cooled donuts into the glaze, allowing excess to drip off. Set glazed donuts on a wire rack and let stand until the glaze sets, about 10–15 minutes.