Add at least 1 inch of vegetable oil to a large heavy skillet (I recommend a 12-inch cast iron skillet). Bring the oil to 360 degrees F over medium heat.
While the oil is heating, make the dough. Place the butter, milk, water, and salt in a medium saucepan and bring to a boil over medium-high heat. Add all of the flour at once, reduce heat to medium-low and stir with a wooden spoon until the mixture forms a ball and appears to dry out (a film or “crust” will develop on the sides and bottom of the pan), another 2 to 3 minutes.
Immediately place the dough into a mixing bowl and beat on low until it stops steaming and is just warm to the touch, approximately 1 minute.
Increase mixer speed to medium and add the egg, beating until fully incorporated and scraping the sides and bottom of the bowl as needed. Add the vanilla extract and mix until thoroughly combined, about 1 minute. The final dough will appear smooth, creamy, and shiny.
Transfer the dough to 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.
Mix together the sugar and cinnamon in a shallow bowl and set aside.
Once the oil comes to temperature, pipe the churros directly into the oil, about 6 inches in length. Use a pair of kitchen shears to snip off the dough at the end of the piping tip. Repeat with additional churros in the oil (don’t overcrowd the pan - you should fit about 4 or 5 per batch). Turn the churros in the oil and cook until golden brown all over, about 2 to 3 minutes. Remove to the paper towel-lined pan to drain for about 15 seconds, or until able to be handled, then toss in the cinnamon-sugar mixture and transfer to a serving dish.
Allow the oil to back up to temperature and repeat step #7 with the remaining dough, approximately three more batches. Serve immediately!
Storing- Cool completely before storing. Keep fried and coated churros in an airtight container lined with paper towels for up to two days.
Freezing- Place cooked and cooled churros on a baking sheet and pre-freeze for 1 hour. Transfer to an airtight container and freeze for up to 1 month for ultimate freshness.
Making ahead- If you plan to make these in advance prep the dough and pipe the churros onto parchment paper. Freeze the dough then transfer the frozen churros to a freezer-safe container for up to 1 month.
Cooking frozen churros- No need to thaw the dough, simply cook the churros following the same frying instructions.