Make the churro donuts: Place the water, milk, butter, sugar, and salt in a medium saucepan and bring to a boil over medium-high heat. Once boiling, remove the pan from the heat and add the flour, stirring with a wooden spoon until the flour is completely incorporated.
Place the pan back over medium heat and continue to cook and stir for 1 to 2 minutes to steam away as much water as possible. The mixture is ready when a thin film coats the bottom of the pan.
Move the mixture to the bowl of a stand mixer fitted with the paddle attachment. Add the vanilla bean paste and mix for about 1 to 2 minutes to allow it to cool. Then mix on medium speed and add the first egg. Let it mix in completely and then scrape down the sides of the bowl.
Add the remaining eggs, one at a time, and mix in completely until the batter becomes smooth and glossy. Transfer the batter to a piping bag fitted with a large open star tip.
Pipe circles onto a parchment paper-lined baking sheet. Top with the desired amount of chocolate chips and then pipe another circle of batter over the top of the chocolate. While the size of the donuts doesn’t matter, keep in mind you need enough batter to cover all of the chocolate; make sure there are no gaps in between each layer otherwise, the melted chocolate will leak out & you’ll make a mess when frying.
Immediately transfer the tray to the freezer and freeze until solid, about 2 hours or overnight.
In a small bowl, whisk together the sugar and cinnamon until combined. Set aside.
Fry the churro donuts: Heat at least 2 inches of oil in a heavy-bottomed pot until a thermometer registers 360-370°F (180-190°C). Working in batches, place two donuts at a time into the oil. Fry on each side until golden brown, about 3 to 4 minutes on each side. Remove fried donuts with tongs and drain on a paper towel, then toss in cinnamon-sugar mixture. Repeat with the rest of the piped churro donuts.