Preheat the oven to 300 degrees.
Pour 2 cups sugar into the center of a deep saucepan. Carefully pour the water around the sugar, 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, 5 to 10 minutes. Immediately remove from the heat. Meanwhile, set a 9-inch cake pan nearby. When the caramel is cooked, quickly pour into the cake pan and swirl to coat the sides. Set aside to cool.
In a saucepan, bring the milk and vanilla bean to a boil over medium heat. Immediately turn off the heat and set aside to infuse. Meanwhile, in a large bowl, whisk together the eggs, additional egg yolks, and the remaining 1 cup sugar. Whisk about 1/2 cup of the hot milk into the egg mixture. Whisk the remaining hot milk into the egg mixture. Strain the mixture into a pitcher to smooth it and to remove the vanilla. Pour the mixture into the caramel-lined cake pan and place in a hot water bath. Bake in the center of the oven until dry and set in the center, 45 to 50 minutes. Remove from the water bath and let cool. Refrigerate at least 4 hours. (Or refrigerate until ready to serve, up to 24 hours.) Turn out onto a serving platter.