• ¾ cup (6 fl oz/180 ml) whole milk
• 3 tablespoons unsalted butter
• 3 cups (15 oz/470 g) all-purpose flour
• ¼ cup (¾ oz/20 g) unsweetened cocoa powder, preferably Dutch process
• ⅓ cup (3 oz/90 g) granulated sugar
• ½ teaspoon salt
• 1 package (2½ teaspoons) quick-rise yeast
• 2 large eggs
• ½ teaspoon vanilla extract
• Canola or peanut oil for brushing and deep-frying
• Chocolate Ganache

In a small saucepan over medium heat, combine the milk and butter and heat, stirring, until the butter melts and the mixture is hot but not boiling (about 125°F/52°C on an instant-read thermometer). Remove from the heat. Set aside.

Fit a stand mixer with the paddle attachment. In the mixing bowl, combine 2½ cups (12½ oz/390 g) of the flour, the cocoa powder, sugar, salt, and yeast and beat on low speed to mix. Add the hot milk mixture, raise the speed to medium, and beat until well blended. Add the eggs and vanilla and beat until incorporated, about 2 minutes. Add the remaining ½ cup (2½ oz/7.5 g) flour and beat until well blended and smooth, about 1 minute longer. The dough will not pull away from the sides of the bowl and will still be somewhat sticky, but your finger should come out clean when you insert it into the center. (For a food processor method.)

Scrape the dough into a large bowl and cover with a clean kitchen towel. Let the dough stand until well risen and increased in bulk (it may almost double in size), about 45 minutes.

Line a baking sheet with waxed paper and brush the paper with oil. Line a second baking sheet with paper towels.

On a generously floured work surface, roll out the dough into a circle 10 inches (25 cm) in diameter and ½ inch (12 mm) thick. Using a 3-inch (7.5-cm) round donut cutter, cut out as many donuts as possible.

Transfer the donuts and holes to the oiled paper. Gather up the donut scraps and repeat rolling and cutting. (For more information on rolling and cutting.) Cover the donuts and holes with a clean kitchen towel and let rise until soft and puffy, about 30 minutes.

Pour oil to a depth of 2 inches (5 cm) into a deep-fryer or deep, heavy sauté pan and warm over medium-high heat until it reads 360°F (182°C) on a deep-frying thermometer. Carefully lower 2–5 donuts or holes into the hot oil and deep-fry until dark brown and crusty on the first side, about 1½ minutes. Turn over and cook until dark brown and crusty on the second side, about 1 minute longer. Transfer to the towel-lined baking sheet. Repeat to fry the remaining donuts and holes, allowing the oil to return to 360°F between batches. (For more information on deep-frying.)

If the ganache seems too thick for dipping, transfer it to a stainless-steel bowl and set over (but not touching) a saucepan of simmering water. Heat, stirring until smooth. When the donuts and donut holes are cool enough to handle, dip the tops in the ganache, letting any excess drip back into the bowl. Place on a wire rack and let stand until the ganache sets slightly, about 10 minutes, then dip a second time. Serve right away.