Step 1

In a small pot, use a fork to combine sugar and 2 tablespoons water to form wet sand. Cook on medium, swirling pot as necessary, until amber in color, 4 to 5 minutes. Remove from heat and whisk in butter (it will sputter), then whisk in heavy cream, cocoa powder, and salt to dissolve.

Step 2

Add chocolate and let sit 3 minutes, then stir until melted and smooth (if desired, use an immersion blender to get it extra-smooth and silky).

Step 3

In a medium saucepan, cook cinnamon stick on medium until fragrant, about 2 minutes. Add 1 cup water, olive oil, 1 tablespoon sugar, and salt and bring to a boil. Discard cinnamon stick, then remove from heat, quickly stir in flour with a wooden spoon, and let sit.

Step 4

Heat vegetable oil in a medium saucepan on medium to 350°F on a candy thermometer. Meanwhile, transfer hot churro batter to a large bowl. Using an electric mixer on low, beat 1 minute to cool slightly. Beat in egg (dough will be smooth and shiny). Transfer batter to a pastry bag fitted with a large star tip. Working in 4 batches, squeeze 3- to 4-inch lengths of batter into oil, using scissors to snip off ends. Fry until golden brown, about 4 minutes, then transfer to a wire rack set over a paper towel.

Step 5

In a small bowl, combine ground cinnamon and remaining 1/2 cup sugar. When churros are cool enough to handle, coat in cinnamon sugar. Serve with chocolate-caramel sauce for dipping.