Combine the water, butter, granulated sugar, and salt in a medium saucepan over medium heat. Stir until the butter is fully melted and the mixture comes to a rolling boil.
Remove the pan from heat and add the flour all at once. Stir vigorously with a wooden spoon until the dough comes together into a smooth ball and pulls away from the sides of the pan, about 1 minute. Let the dough cool for 5 minutes.
Add the eggs to the dough one at a time, stirring vigorously after each addition until fully incorporated before adding the next. Stir in the vanilla extract. The dough should be thick, smooth, and pipeable.
Heat about 2 inches of vegetable oil in a heavy-bottomed pot to 360°F (182°C). Transfer the dough into a piping bag fitted with a large open star tip.
Pipe 1 to 2 inch pieces of dough directly over the oil, using scissors to snip each piece off into the hot oil. Fry in batches of 6 to 8 for about 3 to 4 minutes, turning occasionally, until deep golden brown all over. Transfer to a paper towel-lined plate.
Mix the sugar and cinnamon together in a shallow bowl. While the churro bites are still warm, toss them in the cinnamon sugar until evenly coated.
For the chocolate sauce, place the chocolate chips in a heatproof bowl. Heat the heavy cream in a small saucepan until it just begins to simmer, then pour it over the chocolate chips. Let it sit for 1 minute, then stir until smooth. Stir in the butter until glossy.
Serve the churro bites warm with the chocolate dipping sauce on the side.