To a saucepan over medium-high heat, add the water, unsalted butter, sugar, and salt. Stir until the butter has fully melted and bring to a gentle boil.
Reduce the heat to low, add in the flour, and stir quickly until a smooth dough forms. Continue cooking the dough for about 1 minute (see note 1), stirring constantly, until it pulls away from the sides of the pan and leaves a light film on the bottom. Remove from the heat and let cool for 3–5 minutes.
Add in the eggs one at a time, mixing well with a spatula after each addition until fully incorporated. The mixture may look separated at first, but continue mixing until it comes together into a smooth dough.
Add in the vanilla and mix until the dough is thick, smooth, and pipeable.
Transfer the dough to a piping bag fitted with a large star-shaped piping tip, ideally a closed star tip with about a 1/2-inch opening (see note 2).
In a deep pot, heat vegetable oil to 190°C (375°F), making sure the oil is about 2 inches deep (see note 3).
Carefully pipe the dough directly into the hot oil, keeping the tip close to the surface to prevent splashing. Pipe churros about 6 inches in length, then snip the dough with scissors close to the oil. Fry 3 churros at a time for 4–5 minutes, turning as needed, until deep golden and crisp.
Use tongs to remove the churros from the oil and let them drain briefly, then roll them in the cinnamon sugar while still hot so it sticks well. Repeat with the remaining dough, then serve immediately with the chocolate sauce.