Prepare the Churros: In a medium saucepan, combine water, pumpkin puree, butter, granulated sugar, salt, cinnamon, and vanilla extract. Bring to a boil over medium heat. Remove from heat and add the flour all at once. Stir vigorously until the mixture forms a ball and pulls away from the sides of the pan. Let the mixture cool for 5 minutes. Add the eggs one at a time, beating well after each addition until the dough is smooth and glossy.
Heat the Oil: In a large, deep skillet or pot, heat about 2 inches of vegetable oil to 350°F (175°C).
Pipe the Churros: Transfer the dough to a piping bag fitted with a large star tip. Pipe 4-inch strips of dough into the hot oil, cutting with scissors. Fry the churros in batches until golden brown, about 2-3 minutes per side. Use a slotted spoon to transfer the churros to a paper towel-lined plate to drain.
Coat the Churros: In a shallow bowl, mix together the granulated sugar and ground cinnamon. While the churros are still warm, roll them in the cinnamon-sugar mixture to coat.
Prepare the Chocolate Dipping Sauce: In a small saucepan, heat the heavy cream over medium heat until it begins to simmer. Remove from heat and add the chopped dark chocolate. Let sit for 2 minutes, then stir until smooth and melted. Stir in the vanilla extract.
Serve: Arrange the churros on a serving platter with the bowl of chocolate dipping sauce alongside. Serve warm and enjoy!