For the churros: Combine the butter, salt and 1 1/2 cups water in a medium pot. Bring to a low boil over medium heat. Gradually whisk in the flour, stirring constantly to incorporate. Switch to a wooden spoon and stir over medium heat until the batter is pasty and stiff and all the floury 'lumps' have been worked out, 7 to 9 minutes.
Transfer the batter to the bowl of a stand mixer with the dough hook attachment. Mix together the olive oil and eggs in a separate bowl. Turn the mixer on low and slowly add the oil and eggs to the batter. Increase the speed to medium and mix until smooth, 6 minutes. Remove the batter, cover and chill for at least 1 hour and up to 24 hours.
For the chocolate sauce: Bring the milk to a low simmer in a medium pot over medium heat. Stir in the red pepper flakes.
Thoroughly mix together the cream and cornstarch in a medium bowl or large measuring cup. Pour in a little of the hot milk and mix to temper the cream and cornstarch mixture, and then pour into the pot with the remaining milk. Bring the milk back to a simmer, stirring regularly, until thickened, 7 to 10 minutes. The mixture will require more stirring at the beginning than at the end.
Mix in the chocolate, sugar and cocoa powder. Remove the pot from the heat and let stand for about 10 minutes.
Whisk until smooth and thick. Stir in the rum. The sauce should not be very sweet. It should be thick enough - when hot - to coat the back of a spoon.
For cooking the churros: Add enough canola oil to fill a large pot one-third full. Place over medium heat and heat the oil to 350 degrees F.
Scoop the churro batter into a piping bag fitted with a wide, fluted tip. Pipe out four 6-inch lengths of batter directly into the oil. If it helps, you can pinch off the end of the churro with your fingers and gently drop the length of the churro dough into the oil. Fry, making sure they cook evenly on all sides, until brown and crispy, 6 to 8 minutes. Repeat with the remaining batter.
Meanwhile, mix together the sugar and cinnamon in a baking dish until blended. Add the cooked churros directly to the cinnamon-sugar and gently turn them so they get completely coated on all sides. Tap off the excess cinnamon sugar and transfer the hot, sugar-coated churros to a plate. Serve with a side of the chocolate sauce.
This recipe was provided by a chef, restaurant or culinary professional and may have been scaled down from a bulk recipe. The Food Network Kitchens chefs have not tested this recipe, in the proportions indicated, and therefore, we cannot make any representation as to the results.