Fill a heavy-bottomed pot with 1½–2 inches of oil and heat over medium-high heat to about 360°F.
To make the topping, whisk together sugar and cinnamon in a shallow dish and set aside.
To make the churros, combine water, butter, cinnamon, and salt in a small saucepan. Bring the mixture to a rolling boil over medium heat, then reduce the heat to low.
Add flour and stir to form a ball. Remove from the heat and let cool for 5 minutes.
Add eggs one at a time, stirring until combined after each addition. Add vanilla.
Add a large star tip to a piping bag and spoon dough into the piping bag. Pipe dough over the hot oil and cut at 1–3 inches long. Repeat until churros fill the pan, making sure to leave room between churros. Fry until golden brown, 2–3 minutes on each side.
Remove churros from oil with a slotted spoon and place on a paper towel–lined plate.
While still warm, roll in cinnamon/sugar mixture. Repeat with remaining dough. Serve warm.