Pour milk and water into a microwave safe container. Heat for 20- 30 seconds until warm. Temperature should be between 100 - 110 degrees F. Stir in sugar and active dry yeast. Let sit for 5-8 minutes until mixture is foamy.
Pour mixture into the bowl of stand mixer fitted with a paddle attachment. Add melted butter and lightly whisked eggs. With the mixer running on low speed, slowly add about half the flour. Change to a dough hook. Increase speed and continue to add remaining flour. Continue to knead until mixture comes together to form a smooth dough.
Transfer bowl to a lightly oiled large bowl. Cover with plastic wrap and allow to rest for one hour.
Punch down dough. Place dough on a lightly floured surface. Roll or pat dough to about 1/2-inch thick. Use a round cutter to punch out rounds. Gather any remaining scrapes and cut out smaller rounds. Place cutouts on a parchment lined baking sheet. Cover with plastic wrap as you heat up oil.
Fill a large pot with about 3 inches of vegetable oil. Using a candy thermometer, bring oil to 350 degrees F over medium heat.
Brush off any excess flour off doughnuts before frying. Depending on the size of your pot, add 3-5 doughnuts to the pot at a time. Do not overcrowd the doughnuts, as it will decrease the oil temperature too much. Set a timer for 75 seconds. By this time, the doughnut should have floated to the top. Turn doughnuts over and cook other side for another 60-75 seconds until golden brown all over.* Remove doughnuts from oil and let drain on paper towels. Repeat with smaller doughnut rounds, reducing time to 60 seconds per side. Let cool completely.
Coffee Pastry Cream:
Line a quarter-sheet baking pan with plastic wrap, making sure the sides of the pan are also covered. Set aside.
Pour 1 1/2 cups milk into a large heavy bottomed sauce pot. Add salt and 1/2 cup sugar. Whisk together. Place over medium-high heat. Stir occasionally to make sure the milk does not burn along the sides or bottom of the pan. Bring to a boil.
Meanwhile, in a medium bowl, whisk together cornstarch, 1/4 cup sugar, and 1 Tbsp espresso powder. Add in remaining 1/2 cup milk and yolks. Whisk until smooth.
Once milk has boiled, reduce heat. Slowly add a little bit of the milk into the cornstarch mixture. Whisk constantly to distribute the heat. Continue to add milk and whisk until only a couple tablespoons of milk are left in the saucepan. Pour all the egg-milk mixture back into the saucepan.
Whisk constantly over medium-low heat until the custard has thickened. Keep whisking and let the custard come to boil to cook out the cornstarch flavor. Remove from heat. Whisk in butter until smooth. Add vanilla and whisk until combined.
Pour the hot pastry cream into the prepared plastic wrapped sheet pan. Spread evenly. Cover with plastic wrap and freeze for 10-15 minutes until the pastry cream has cooled to room temperature. Remove from freezer and transfer to an airtight container. Place plastic wrap directly on pastry cream to avoid condensation. Let chill in the refrigerator until ready to use.
Cinnamon Sugar and Assembly:
Once doughnuts are cool, use a chopstick to poke a hole of the side of each doughnut. Fit a pipping bag with a small round tip. Transfer cooled coffee pastry cream to pipping bag. Fill doughnuts with coffee pastry cream.
In a small bowl, whisk together sugar and cinnamon. Keep melted butter in a separate bowl.
Lightly brush cooled doughnuts with butter. Dunk doughnuts in cinnamon sugar mixture. Serve and enjoy immediately!
* Sacrifice one doughnut as a tester to make sure the doughnuts are cooked throughout. If not, increase cooking time accordingly.