Line a plate with paper towels.
Heat water, sugar, butter, and salt over medium heat in a medium-sized saucepan.
Bring it to a boil, then quickly remove the saucepan from the heat. Whisk in the flour and continue mixing with a whisk or wooden spoon until it forms a soft ball. Let it cool for 2-3 minutes. Leave it in the saucepan or transfer to a bowl if the saucepan isn't large enough.
Add eggs and vanilla to the flour mixture and mix with an electric mixer. Don't be alarmed if the mixture starts to break apart. Keep mixing until smooth.
Heat 3-4 inches of vegetable oil in a large, heavy-bottomed pot over medium-high heat until it reaches 375℉ (190℃).
Put the dough into a piping bag with a star tip. Push the dough down to the tip. (If you don't have a piping bag, put the dough in a ziplock or sandwich bag with a fold-over top and cut off a small part of one on the bottom corners. Push the dough to that corner.)
Holding the piping bag a few inches above the oil, carefully pipe the churro dough into ropes 4-5 inches long directly into the oil. Cut the ropes with kitchen scissors or a sharp knife. Be careful because you don't want the oil to splatter as they fall.
Add 3-4 churros, depending on the size of your pot and the amount of oil. You don't want to overcrowd the pan, or it will lower the oil temperature too much, resulting in greasy churros. Let them fry for about 2 minutes per side until golden brown and cooked through.
Remove the churros and place them on the paper-towel-lined plate to drain. Roll them in the cinnamon sugar. Repeat with the rest of the dough.