Prepare the Dough: In a medium saucepan, combine the water, butter, granulated sugar, and salt. Bring to a boil over medium heat.
Add Flour: Once the mixture starts to boil, reduce heat to low. Add the flour all at once and stir vigorously with a wooden spoon until the dough comes together in a smooth ball and pulls away from the sides of the pan, about 1-2 minutes. Remove from heat and let the dough cool for a few minutes.
Add Egg and Vanilla: Once slightly cooled, add the egg and vanilla extract to the dough. Stir until the egg is fully incorporated, and the dough is smooth and sticky.
Prepare for Frying: Heat about 2 inches (5 cm) of oil in a heavy-bottomed pot over medium heat until it reaches 350°F (175°C). Meanwhile, transfer the dough to a piping bag fitted with a large star tip.
Pipe and Fry Churros: Pipe 4- to 5-inch (10-13 cm) strips of dough into the hot oil, using scissors to cut off each piece. Fry a few churros at a time, making sure not to overcrowd the pan. Fry until golden brown, turning occasionally, about 3-4 minutes per churro.
Drain and Coat: Remove churros with a slotted spoon and place them on a paper towel-lined plate to drain excess oil. Immediately toss the warm churros in the cinnamon-sugar mixture until well coated.