In a saucepan over medium-high heat, melt butter then add water, salt, vanilla, and cinnamon stick. Bring to a boil then remove the cinnamon stick and remove the mixture from heat.
Add the flour and stir with a wooden spoon until a smooth dough is formed, about 3-5 minutes.
Let dough cool, about 10-15 minutes then transfer to a bowl and add the egg. Stir vigorously until egg is incorporated. The mixture is pretty hot so you have to stir vigorously so the egg doesn't actually cook and you get scrambled eggs :)
In a shallow bowl, mix together sugar and cinnamon. Set aside.
Heat oil in a Dutch oven to 400 °F (204 °C)
Hold piping bag above oil and pipe about four 6-inch lengths of dough. Fry until golden brown, about 2 minutes, flipping every so often with a spider strainer.
Transfer to paper towels to drain briefly then transfer to cinnamon-sugar mixture and roll around until evenly coated. Repeat with remaining dough in piping bag.