Place the water, milk, butter, granulated sugar, brown sugar, salt and cinnamon into a medium sauce pan. Stir over medium heat until mixture comes to a boil.
Reduce heat to low, add the flour, and stir vigorously until smooth. The dough should pull away from the sides of the saucepan and form a ball.
Transfer dough to a stand mixer fitted with the paddle attachment. Allow it to cool for 5 minutes.
Add the vanilla and one egg, mixing until combined. Add the second egg and continue mixing until the dough is smooth and creamy.*
Transfer dough to a large pastry bag fitted with a large star tip (I used Wilton's 1M tip).**
Pour the oil into a deep frying pan and heat it to 360ºF. The oil should be 1.5-2 inches deep.
While the oil is heating up, line a rimmed baking sheet with two layers of paper towels.
Make the Cinnamon-Sugar Coating by combining the granulated sugar and cinnamon in a shallow dish.
Once the oil reaches the right temperature, pipe the churros directly into the oil, about 5-6 inches in length. Use a pair of kitchen shears (sprayed with nonstick cooking spray) to snip the dough from the pastry tip. Fry in batches of 4 churros at a time. Try not to overcrowd the pan.
Fry in oil for 2 minutes, then turn and fry for another 2 minutes, or until golden on all sides.
Remove churros from oil, placing them onto the paper towel-lined baking sheet to soak up any excess oil. Allow to cool 20-30 seconds before tossing them into the cinnamon-sugar coating.***