Instructions

for the doughnuts

Stir flour, sugar, and yeast together in bowl of stand mixer. Add milk and egg and mix with rubber spatula until all ingredients are moistened. Fit mixer with dough hook and mix on medium-low speed until cohesive mass forms, about 2 minutes, scraping down bowl if necessary. Cover bowl with plastic wrap and let stand for 20 minutes.
Add salt and mix on medium-low speed until dough is smooth and elastic and clears sides of bowl, 5 to 7 minutes. With mixer running, add butter, a few pieces at a time, and continue to mix until butter is fully incorporated and dough is smooth and elastic and clears sides of bowl, 7 to 13 minutes longer, scraping down bowl halfway through mixing. Transfer dough to lightly greased large bowl, flip dough, and form into ball. Cover bowl with plastic. Let sit at room temperature for 1 hour. Transfer to refrigerator and chill overnight (or up to 48 hours).

for the pastry cream

Heat half-and-half in medium saucepan over medium heat until just simmering. Meanwhile, whisk egg yolks, sugar, and salt in medium bowl until smooth. Add flour to yolk mixture and whisk until incorporated. Remove saucepan from heat and, whisking constantly, slowly add ½ cup warm half-and-half to yolk mixture to temper. Whisking constantly, return tempered yolk mixture to half-and-half in saucepan.
Return saucepan to medium heat and cook, whisking constantly, until mixture thickens slightly, about 1 minute. Reduce heat to medium-low and continue to simmer, whisking constantly, for 8 minutes longer. Increase heat to medium and cook, whisking vigorously, until bubbles burst all over surface, 1 to 2 minutes. Off heat, whisk in butter and vanilla until butter is melted and incorporated. Strain pastry cream through fine-mesh strainer set over medium bowl. Press lightly greased parchment paper directly on surface and refrigerate until set, at least 2 hours or up to 24 hours.
Adjust oven racks to lowest and middle positions. Place loaf pan on lower rack. Line rimmed baking sheet with parchment and grease parchment. Transfer dough to lightly floured counter. Press into 8-inch square of even thickness, expelling as much air as possible. Roll dough into 10 by 13-inch rectangle, about ½ inch thick. Using 3-inch round cutter dipped in flour, cut 12 rounds. Transfer doughnuts to prepared sheet. Bring kettle or small saucepan of water to boil.
Pour 1 cup boiling water into loaf pan. Place sheet on upper rack, uncovered. Close oven and allow doughnuts to rise until dough increases in height by 50 percent and springs back very slowly when pressed with your knuckle, about 1 hour.
About 20 minutes before end of rising time, add oil to large Dutch oven until it measures about 1½ inches deep and heat over medium-low heat to 330 degrees. Set wire rack in second rimmed baking sheet and line with triple layer of paper towels. Using both your hands, gently place 4 risen doughnuts in oil. Cook until golden brown on undersides, 1½ to 2 minutes, adjusting burner as necessary to maintain oil temperature between 325 and 340 degrees. Using spider skimmer, flip doughnuts and cook until second sides are browned, 1½ to 2 minutes. Transfer doughnuts to prepared rack. Return oil to 330 degrees and repeat with remaining doughnuts. Let doughnuts cool completely, about 20 minutes.

for the chocolate frosting

Microwave chocolate and water in medium bowl at 50 percent power until melted, about 30 seconds. Whisk in sugar, cocoa, and salt until smooth and fluid. Set aside and let cool slightly. Meanwhile, spoon pastry cream into pastry bag or zipper-lock bag fitted with ¼-inch round pastry tip.
Insert paring knife through side of 1 doughnut until tip almost reaches opposite edge. Swing knife through doughnut, creating large pocket. Repeat with remaining doughnuts. Stand doughnuts slit side up in 13 by 9-inch baking pan. Remove paper towels from wire rack and discard (do not wash rack or sheet).
To fill doughnuts, insert pastry tip ¾ inch into opening and squeeze gently until cream just starts to appear around opening, about 2 tablespoons cream per doughnut. (Reserve any remaining pastry cream for another use.)
If frosting stiffens before using, microwave it at 50 percent power, stirring every 30 seconds, until smooth and fluid. Dip top half of 1 doughnut into frosting until evenly coated, allowing excess to drip back into bowl. Invert and place on unlined wire rack. Repeat with remaining doughnuts. Let stand until frosting has become slightly matte and dry to touch, 15 to 30 minutes, before serving.