Line a baking sheet with parchment paper.
In a large saucepan over medium-high heat, melt butter.
Add in water, salt, vanilla, and cinnamon stick.
Bring to a boil, remove from heat, and remove cinnamon stick.
Add in the flour and stir until combined well and a dough is formed. Approximately 3-5 minutes.
Let mixture cool and then add in pumpkin puree and egg and mix well.
Transfer dough to a piping bag fitted with a large star tip. Set aside.
In a shallow dish, mix together cinnamon and sugar.
Pipe 6 inch long dough logs onto prepared baking sheet. Flash freeze for 10 minutes.
In a dutch oven or large skillet, heat oil to 400°.
Remove churros from freezer and place frozen dough into the hot oil.
Fry until golden brown, about 3-5 minutes, flipping often.
Remove from oil with tongs and place on paper towel, then immediately roll into cinnamon sugar.
Repeat with the rest of the dough.