Stir the sugar and 2 tablespoons of the water together in a heavy-bottomed saucepan. Cover and bring to a boil over medium heat. Uncover and continue to cook, swirling the pan but not stirring, until the sugar is a dark golden caramel, 7 to 10 minutes. (It should smell slightly burnt.) Pull pan from the heat and carefully pour the 1/4 cup water into the caramel. Take care; it may spatter a lot.
Whisk until smooth and allow to cool. (The burnt caramel can be held for up to 2 weeks at room temperature.)
When ready to serve: Put 3 tablespoons of the burnt caramel into 4 mugs or cups and top evenly with the chopped chocolate. Bring the milk to a simmer in a medium pan over medium heat. Pull pan from the heat and add the Goldschlager and rum; pour over the chocolate.
Serve El Dorado Hot Chocolates with a spoon to stir the layers together.
