Stir together milk and yeast in the bowl of an electric mixer. Let stand until foamy, about 5 minutes.
Attach bowl to mixer fitted with the dough hook. Add egg, sugar, flours, and salt; mix on medium speed until dough comes together, about 6 minutes.
Add butter and 1 teaspoon espresso powder; mix on medium-high speed until combined, about 3 minutes. Scrape down sides of bowl. Raise speed to high, and mix until dough forms a smooth ball, about 1 minute more.
Reduce speed to low. Add remaining 5 teaspoons espresso powder, and mix until just combined. Shape dough into a smooth ball; cover with plastic wrap. Let dough rise in a warm, draft-free spot until doubled in bulk, 30 to 40 minutes.
On a lightly floured surface, roll out dough to 1/2 inch thick. Cut out rounds with a 2 1/2-inch round cutter. Using a 3/4-inch round cutter, cut out center of each round. Gather together scraps, reroll, and cut out. Transfer to a baking sheet lined with parchment paper, and cover with a dry kitchen towel. Let rise until almost doubled in bulk, 20 to 25 minutes.
Heat about 5 inches of oil in a large, heavy stockpot until it registers 360 degrees.on a deep-fry thermometer. Working in batches of 4 to 6, fry doughnuts, flipping once, until golden brown, 1 to 1 1/2 minutes per side. Using a slotted spoon or kitchen tongs, transfer to a wire rack set over a baking sheet lined with paper towels. (Adjust heat between batches as necessary to keep oil at a steady temperature.) Let cool 10 minutes before glazing.
Whisk glaze. Dip each warm doughnut into glaze, turning to coat completely. Transfer to a wire rack set over a baking sheet lined with parchment; let glaze set, about 20 minutes. Place in a single layer on a baking sheet lined with parchment until ready to serve (up to 8 hours).
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