Start by heating at least 1 inch of vegetable oil in a large heavy skillet. I recommend using a 12-inch cast iron skillet for even heat distribution. Bring the oil temperature to 360 degrees F over medium heat.
While the oil is heating, prepare the dough. In a medium saucepan, combine the butter, milk, water, and salt. Bring this mixture to a boil over medium-high heat, stirring occasionally.
Once it reaches a boil, add all of the flour at once. Reduce the heat to medium-low, and stir the mixture vigorously with a wooden spoon. Continue stirring until the dough forms a ball and starts to pull away from the sides of the pan. This should take about 2 to 3 minutes.
Transfer the dough into a mixing bowl. Beat it on low until it stops steaming and is just warm to the touch, which should take about 1 minute.
Gradually increase the mixer speed to medium, and add the egg, beating until fully incorporated. Be sure to scrape down the sides and bottom of the bowl as needed. Mix in the vanilla extract until everything is well combined and you achieve a smooth, creamy, and shiny dough.
Prepare a pastry bag fitted with an extra-large open star piping tip (I recommend Ateco #829). Line a rimmed baking sheet with two layers of paper towels to absorb excess oil.
In a shallow bowl, mix together the sugar and cinnamon. Set this mixture aside for later.
Once the oil has reached the right temperature, carefully pipe the churros directly into the oil, aiming for about 6 inches in length. Use kitchen shears to snip off the dough at the end of the piping tip. Be careful not to overcrowd the pan; you should be able to fit about 4 or 5 churros at a time.
Fry the churros until they’re golden brown all over, which should take about 2 to 3 minutes. Use tongs to turn them in the oil for even cooking. When they’re done, remove them to the paper towel-lined pan to drain for about 15 seconds or until they can be handled.
Immediately toss the churros in the cinnamon-sugar mixture and transfer them to a serving dish. Allow the oil to return to temperature, then repeat the frying process with the remaining dough, which should take about three more batches. Serve the churros warm!