Add the sugar and water to a small saucepan. Cook over low heat without stirring with a spoon. If needed, gently swirl the pan until the sugar dissolves and begins turning golden.
Once the caramel starts changing from golden to amber, remove it from the heat. Do not wait until it becomes dark brown because the residual heat will continue cooking it.
Carefully pour the hot caramel into the pan and swirl it across the bottom and slightly up the sides. Use an oven mitt or kitchen towel because the pan will become very hot. Let the caramel coo
Blend the Quesillo:
Add the whole milk, condensed milk, eggs, vanilla, and rum to a blender. Blend for about 40 seconds, or just until smooth.
For a smoother quesillo, pour the mixture through a fine-mesh strainer into the caramelized pan. This removes the foam and some of the air bubbles.
Cover the pan with aluminum foil and place it inside the larger baking dish. Pour hot water into the outer dish to create a bain marie.
Bake the Quesillo:
Bake for about 1 hour to 1 hour 10 minutes, or until a knife or toothpick inserted into the center comes out clean.
Let the quesillo cool completely at room temperature, then refrigerate for at least 6 to 8 hours. Overnight is even better.
Unmold
Run a thin knife or small spatula around the edges. Place a serving plate over the pan and carefully flip it over. Lift the pan and let the caramel run over the top.