Pour 2 cups of the sugar into the center of a deep saucepan and carefully pour the water around it, trying not to splash any sugar onto the sides of the pan. Do not stir; gently draw your finger through the center of the sugar twice (making a cross) to moisten it. Over medium-high heat, bring to a boil without stirring. Reduce the heat to a fast simmer and cook without stirring until amber-caramel in color, 10 to 20 minutes. Immediately remove from the heat. Set 8 (1-cup) ramekins or shallow coffee cups nearby. When the caramel is cooked, quickly pour about 1/4 cup of caramel into each ramekin and swirl to coat the sides. Set aside to cool. Preheat the oven to 325 degrees F.

In a saucepan, bring the milk, vanilla bean and cinnamon sticks to a boil over medium heat. Immediately turn off the heat and set aside to infuse for 10 minutes. Add the chopped chocolate and whisk to melt and combine. Meanwhile, whisk together the eggs, additional egg yolks, and the remaining cup of sugar. Whisk about 1/2 cup of the hot milk into the egg mixture and then whisk in the remaining hot milk. Strain the mixture into a pitcher to smooth it and to remove the vanilla bean and the cinnamon sticks.

Pour the mixture into the caramel-lined ramekins and arrange in a hot water bath. Bake in the center of the oven until set in the center, 30 to 35 minutes. Remove from the water bath and let cool. Tightly cover each ramekin with plastic wrap. Refrigerate at least 2 hours. (Or refrigerate until ready to serve, up to 24 hours.)

To serve, turn out flans onto plates (you may need to dip the bottom of each ramekin in hot water to loosen the caramel.) The caramel will pour out and serve as the sauce.