Combine the yeast and water in a medium bowl. Microwave the milk and butter on a low setting until the milk is warm and butter is just melted, about 45 seconds to 1 minute. Be sure not to overheat the mixture.
In a separate bowl whisk together the egg and sugar until the sugar is dissolved. Add the warm butter and milk to the egg mixture. Whisk to combine then pour it into the yeast.
In another bowl, mix together the flour, salt and nutmeg. With a wooden spoon, add the flour mixture to the wet ingredients. Mix until all the ingredients are smooth and well combined into soft dough 1 to 2 minutes. Cover the bowl with a kitchen towel let sit in a warm place until it doubles in size, about 1 1/2 to 2 hours.
On a lightly floured work surface gently roll the dough into a 1/2-inch thick rectangle (about 10 by 18 inches). Cut the dough in 32 smaller rectangles, each about 1 by 1 1/2 inches. Sprinkle flour on 2 baking sheets to prevent the dough from sticking. Space the rectangles of dough about 2 inches apart on the baking sheets. Cover and let rise in a warm place until doubled in size, about 2 hours.
Preheat the oil in fryer to 375 degrees F.
Fry the doughnuts in the oil until golden brown, about 1 minute on each side, turning with tongs. Remove the doughnuts from the oil and place on a cooling rack, allowing some of oil to drain on a paper shopping bag or paper towels placed below rack.
Once doughnuts have cooled to room temperature sift confectioners' sugar over the top of the doughnuts. Move the doughnuts to a serving platter and sprinkle the tops of each one with 1/4 teaspoon of espresso. Serve warm or at room temperature.
Chef's notes: These seem like they take a while to make but, really, all of the time is spent waiting for the dough to rise.
A great way to serve these is to place a doughnut in top of a steaming cup of fresh-brewed coffee.