Boil. Heat water, butter, sugar, vanilla, and salt in a medium saucepan over medium-high heat. Bring to a boil.
Add flour. Remove from heat, add flour and beat 1 minute or until completely combined.
Cool. Transfer mixture to a stand mixer fitted with a paddle attachment. Let cool 5-10 minutes.
Beat in egg. Add egg and beat on medium until fully incorporated and a soft dough has formed.
Pipe. Transfer to a pasty bag fitted with a 1/2 inch open star tip. Pipe onto a parchment-lined baking sheet into long sticks, spirals, or any shape you'd like.
Freeze. Stick the baking sheet in the freezer until churros are stiff, about 20-30 minutes.
Heat oil. Meanwhile, pour vegetable oil into a wide, shallow frying pan or braiser. Attach a candy or deep fry thermometer to the side and heat to 360°F over high heat. Monitor the heat as you are frying the churros, you don't want it to get below 340°F.
Fry. Once churros are stiff and oil is hot, fry the churros 3 or 4 at a time until golden on all sides, about 4 minutes total. If the churros are too floppy when you try to flip them give them another minute to cook, then try to flip.
Drain. Transfer to a paper towel-lined baking sheet and let drain for a minute or two.
Coat in cinnamon-sugar. Combine sugar and cinnamon for coating in a shallow bowl or pie plate. Place the warm churros in the cinnamon mixture and coat on all sides.
Serve. Serve while still warm with Mexican chocolate sauce.