In a large saucepan, combine the water, butter, sugar, and salt. Bring to a rolling boil over medium-high heat, then promptly remove from the burner. Immediately stir in the flour using a wooden spoon, continuing vigorously until the mixture is fully incorporated and forms a cohesive mass, about 30 to 60 seconds. Return the pan to the heat and cook, stirring constantly for 30 seconds. Transfer the dough into a stand mixer fitted with a paddle attachment or use a hand mixer. Begin mixing at medium speed, incorporating the eggs one at a time along with the vanilla extract. Pause after each egg to scrape down the bowl, mixing until the dough becomes smooth, glossy, and thick enough to slowly fall from the beaters. If it remains too firm, incorporate the optional third egg fully.

Heat the vegetable oil to 370°F (188°C), ideally measured with a candy thermometer. Using a piping bag fitted with a large closed star tip (such as an 869), pipe out 6-inch lengths of dough directly into the oil, working in small batches of about three. Fry each churro for 90 seconds to 2 minutes per side, turning once, until evenly golden. Use tongs to remove them, draining briefly on paper towels before coating in cinnamon sugar. Generously spoon sugar over the hot churros, shake off the excess, and serve immediately.

To prepare the chocolate sauce, place chocolate chips, cinnamon, and heavy cream in a heatproof bowl set over a pot of gently simmering water. Stir occasionally until the mixture is fully melted and homogenous.