In a small bowl, dissolve yeast in warm water for about 5 minutes, until frothy.
In the bowl of a mixer or in a large bowl with a hand mixer, whisk together flour, espresso powder, sugar, and salt.
Add in milk, butter, egg yolks, and yeast mixture and blend together using the mixer’s dough hook (if you don’t have a mixer with dough hook, a hand mixer is fine). Form dough and then increase mixer speed to medium and let mix for about 3 minutes.
Sprinkle the dough with some flour, cover with a kitchen towel, and place in a warm place for about two hours, so it can rise.
Place dough on a lightly floured surface and roll out so it’s about ½ inch thick. Using a biscuit cutter or the top of a drinking glass (or other round object), cut out as many rounds from the dough as you can. I used a 3-inch cutter and got 12. Place rounds on a lightly floured baking sheet. Cover the baking sheet with your kitchen towel and let rise again for about 30 minutes.
Heat 2.5 inches of canola oil in a large pot until it reaches 350 degrees. Add a few dough rounds to the oil and fry for about a minute on each side until they start turning golden. Place fried doughnuts on a paper towel-covered plate to rest. When finished frying, move doughnuts to a wire rack to cool completely.
Fill a pastry bag with long tip with custard. Pierce each doughnut and squeeze some custard into the middle. Drizzle a layer of icing over the top.