Step 1: In a medium saucepan, combine water, butter, 2 tablespoons granulated sugar, and salt. Bring the mixture to a rolling boil over medium-high heat, stirring occasionally until butter is melted.
Step 2: Remove the saucepan from heat. Add all the flour at once and stir vigorously with a wooden spoon until the mixture forms a smooth ball and pulls away from the sides of the pan.
Step 3: Return the saucepan to low heat and cook the dough for 1-2 minutes, stirring constantly, to dry it out. A thin film should form on the bottom of the pan.
Step 4: Transfer the dough to a large mixing bowl. Let it cool for 5-10 minutes.
Step 5: Add the eggs one at a time to the cooled dough, mixing thoroughly after each addition until fully incorporated and the dough is smooth and glossy.
Step 6: In a shallow dish, combine 1/2 cup granulated sugar and 1 tablespoon ground cinnamon for the churro coating. Set aside.
Step 7: Heat vegetable oil in a large pot or Dutch oven over medium-high heat until it reaches 350-375°F (175-190°C). Use a candy thermometer for accuracy.
Step 8: Transfer the churro dough to a piping bag fitted with a large star tip (like Ateco 826). Carefully pipe 4-6 inch strips of dough directly into the hot oil, cutting with kitchen shears.
Step 9: Fry 3-4 churros at a time for 2-4 minutes per side, or until golden brown and cooked through. Do not overcrowd the pot.
Step 10: While churros fry, prepare the chocolate sauce: gently heat heavy cream in a small saucepan until it just simmers. Remove from heat.
Step 11: Add chopped dark chocolate to the hot cream. Let sit for 2-3 minutes, then stir until smooth. Stir in vanilla extract. Keep warm.
Step 12: Remove fried churros from oil with a slotted spoon and place on a paper-towel-lined plate to drain briefly.
Step 13: Immediately toss the warm churros in the cinnamon sugar mixture until evenly coated.
Step 14: Serve the warm churros with the decadent chocolate dipping sauce.