To make the caramel, heat the sugar and 3 tablespoons of water in a saucepan over medium heat. Refrain from stirring; instead, gently swirl the pan to evenly distribute the heat until the sugar dissolves and the mixture turns golden brown. Exercise caution to avoid burning.
Once caramelized, add the remaining water and mix gently with a spoon. Pour the mixture into the mold for the quesillo, ensuring even coverage of the walls and base. Allow it to cool and harden.
Preheat your oven to 350°F (175°C).
In a blender, combine the eggs, condensed milk, milk, sugar, vanilla extract, and rum (optional). Blend until the mixture is smooth and well combined.
Pour the blended mixture into the prepared caramel-coated mold.
Place the mold in a larger baking dish or roasting pan. Fill the larger dish with hot water until it reaches about halfway up the sides of the mold, creating a water bath.
Carefully transfer the water bath with the mold into the preheated oven. Bake for about 45-60 minutes, or until the quesillo is set around the edges but still slightly jiggly in the center.
Once cooked, remove the quesillo from the oven and allow it to cool completely in the water bath.
Once cooled, carefully remove the mold from the water bath and refrigerate the quesillo for at least 4 hours, or overnight, to allow it to fully set.
To serve, run a knife around the edges of the mold to loosen the quesillo. Place a serving plate on top of the mold and invert it to release the quesillo onto the plate, revealing the caramelized top. Slice and serve chilled.