To prepare the doughnuts:

1. Heat the milk on the stove or in the microwave until lukewarm but not hot. Pour the milk into a large mixing bowl or the bowl of a stand mixer and sprinkle the yeast over top of it. Allow the yeast to dissolve, about 5 minutes.
2. Once the yeast is foaming, add the eggs, melted butter, sugar, salt, and two cups of flour to the mixing bowl. Mix on low speed until combined and then add the remaining flour. Switch to a dough hook attachment and mix on medium-low speed until the dough begins to pull away from the sides of the bowl. If it’s still too wet to pull off the bowl add flour two tablespoons at a time until the dough is tacky and pulling away. Knead for a minute and then place the dough in a lightly greased bowl. Cover tightly with a sheet of plastic wrap and allow the dough to rise in a warm spot of the kitchen for about an hour or until it has doubled in size.
3. Once risen, turn the dough out on a floured surface and roll or pat out until ½ thick. Use a floured doughnut cutter or a 2-3/4-3” drinking glass to cut out rounds of dough. If you don’t want to fill your doughnuts, you can cut out center holes as well, saving the small pieces for doughnuts holes. Knead any scraps together and repeat this process. Place your cut out doughnuts 2” apart on a floured baking sheet and cover with a kitchen towel or greased sheet of plastic wrap to rise in a warm spot of your kitchen a second time. Once the doughnuts are slightly puffed, about 45 minutes, heat the oil in a heavy bottomed pan or fryer to 375 degrees. Combine the cup of sugar with 2 teaspoons of cinnamon in a large baking dish or rimmed sheet pan. Add a few doughnuts to the oil, being careful not to overcrowd the pan or burn yourself with the oil. Cook on one side for about 45 seconds or until golden and then use a metal spatula to carefully flip to the other side. Cook for another 45 seconds or until golden. Remove the doughnuts to a towel-lined baking sheet or cookie sheet to cool for about 30 second to a minute and then carefully shake in the pan of cinnamon and sugar. Repeat this process with the remaining doughnuts, being sure to keep the oil to temperature. Allow to cool prior to filling to custard or cream. When ready to fill, poke a small hole into the side of the doughnuts to be filled and pipe the custard or cream into the doughnut using a piping bag or a large plastic bag with the end snipped off. Pipe until the doughnuts are full. Doughnuts are best eaten the day they are prepared.

To prepare the custard:

1. Combine the milk, ¼ cup sugar, and espresso powder in a medium saucepan and bring to a boil over medium heat. Meanwhile, in a separate large bowl, whisk together the egg yolks, whole egg, 1/3 cup sugar, and cornstarch. Once the milk mixture has begun to boil, carefully remove the pot from the stove and ladle a small stream of the milk mixture into the egg mixture, whisking all the while. Be careful to whisk vigorously the whole time to avoid cooking the eggs and curdling your mixture. Add more of the milk mixture, continuing to whisk, until about half of the milk has been integrated into the eggs. Pour all of the mixture back into the saucepan and cook, whisking constantly, until bubbling and thickened to a pudding consistency. Remove from heat and add the butter and vanilla, stirring to combine, and pour the mixture into a heat-safe bowl. Cover with a sheet of plastic wrap directly touching the custard and refrigerate until the mixture has cooled at least to room temperature.

To prepare the cream:

1. In a small bowl, combine the milk, espresso powder, and vanilla until the espresso powder is mostly dissolved. In a large bowl or the bowl of a stand mixer, cream the butter until light and fluffy. Scrape the sides of the bowl and add the powdered sugar. Stir on low speed until combined and add the milk mixture. Whip to combine, about one minute. Add more sugar if your frosting appears a bit loose, or add more milk if it is too stiff.