Heat about 1 1/2-2 inches of vegetable oil in a large pot or deep cast-iron skillet over medium-high. The oil should reach 360 degrees Fahrenheit, use a candy thermometer to check the temperature.
Carefully pipe the mixture into the preheated oil, pressing about 6-inch lengths of churros and cutting the end with clean kitchen scissors. Do not overcrowd the pan.
Fry until golden brown, about 2 minutes per side.
Use a slotted spoon to transfer the fried churros to paper towels for about 2-3 minutes, after that toss them immediately into the cinnamon-sugar mixture.
Repeat with the remaining dough.
Serve warm with chocolate or caramel sauce for dipping.