Combine the water, butter, 1/4 cup sugar, and salt in a medium saucepan over high heat. Bring to a boil and cook until the butter is completely melted and the sugar is dissolved. Add the flour, 1/4 teaspoon cinnamon and 1/4 teaspoon anise seeds and quickly beat with a wooden spoon until a ball is formed and the mixture becomes dry, about 2 minutes.
Transfer the mixture to the bowl of a stand mixer fitted with the paddle attachment and beat for one minute to allow some of the steam to escape the mixture and to cool slightly before adding the eggs. Add the eggs, one a time, beating every time until each is incorporated. Add the vanilla extract and beat for 20 seconds longer. Scrape the mixture into a pastry bag fitted with a large star tip.
Combine the remaining 1 cup sugar, 1/2 teaspoon cinnamon and 1/2 teaspoon anise in a baking dish.
Working in batches, pipe 6-inch strips of the dough into the hot oil and fry until golden brown, turning once. Remove with a slotted spoon or spider to the prepared paper towel-lined baking sheet. Let the churros drain for 30 seconds and immediately toss in the sugar mixture. Serve the churros warm with the chocolate dipping sauce on the side.