Step 1: Prepare the cinnamon sugar coating and chocolate sauce. In a shallow dish, combine 1/2 cup granulated sugar and 1 tablespoon cinnamon; set aside. For the chocolate sauce, combine chocolate chips, heavy cream, and butter in a microwave-safe bowl. Heat in 30-second intervals, stirring until melted and smooth. Stir in vanilla extract and keep warm.
Step 2: Make the churro dough. In a medium saucepan, combine 1 cup water, 1/2 cup butter, 1/4 teaspoon salt, and 1 tablespoon sugar. Bring to a rolling boil over medium-high heat, stirring until butter is melted. Remove from heat immediately. Add the 1 cup all-purpose flour all at once and stir vigorously with a wooden spoon until a smooth ball of dough forms and pulls away from the sides of the pan.
Step 3: Incorporate the eggs. Transfer the dough to a large mixing bowl or the bowl of a stand mixer fitted with the paddle attachment. Let cool for 5 minutes. Add eggs one at a time, beating well after each addition, until the dough is smooth and glossy. The dough should be thick but pipeable.
Step 4: Heat the oil and pipe the churros. In a large, heavy-bottomed pot or Dutch oven, heat 4-6 cups of vegetable oil to 350-375°F (175-190°C). Use a candy thermometer to monitor the temperature. While the oil heats, transfer the dough to a piping bag fitted with a large star tip (like Ateco 826 or 829). Carefully pipe 4-6 inch strips of dough directly into the hot oil, using kitchen shears or a knife to cut the dough. Do not overcrowd the pot; fry in batches.
Step 5: Fry and coat the churros. Fry the churros for 2-3 minutes per side, or until golden brown and cooked through. Remove with a slotted spoon or spider and place them on a paper towel-lined plate to drain excess oil for a few seconds. Immediately transfer the hot churros to the cinnamon sugar mixture and roll to coat evenly. Serve warm with the chocolate dipping sauce.