Preheat oven to 400° F.
Put the milk, coffee, butter, sugar, and espresso into a 2-quart saucepan and bring to a full boil over medium heat, stirring frequently with a spatula until the butter melts. Still stirring, add the flour all at once. Vigorously stir until the flour is thoroughly incorporated, then continue to cook and stir for another 30 to 45 seconds, or until the dough forms a ball and a light crust is visible on the bottom of the pan.
Remove the pan from the heat and scrape the paste into a medium bowl. Immediately, while the dough is still hot, beat in the eggs one at a time with a mixer using the paddle attachment, on low speed.
After you have incorporated the five eggs, take a good look at the mixture. Continue to beat until you can place the dough between your thumb and index finger and pull apart about an inch without the dough tearing.
Once blended, spoon the choux paste into a pastry bag fitted with a ½ inch plain tip and pipe quarter sized puffs onto parchment lined baking sheets, leaving about 1 inch between puffs. Use a moistened fingertip to poke down and adjust any small imperfections. Brush each of the pastries with a little egg wash.
Bake for 20 minutes; lower the temperature to 350°F and bake 5 to 7 minutes longer until the pastries are golden brown and feel hollow. Halfway through the baking period, rotate the baking sheets. Once completed, transfer sheets to cooling racks and allow the puffs to cool to room temperature before cutting and filling.