Preparation:
To make the caramel sauce to cover the baking dish, mix the sugar and water over high heat in a heavy saucepan. The sugar will start to boil and then will start to brown, from a pale amber into a medium dark color. When it turns to a dark rich color, remove from heat and pour into the baking dish. Tilt the dish back and forth to cover the bottom and sides. Handle carefully; the caramel at this stage will be very hot. Set aside and let the caramel cool.
For the custard, preheat oven to 350 degrees. Pour the can of sweetened condensed milk into a blender and set aside. In a separate bowl, pour the whole milk, add the grated lime, and put the bowl in the microwave for about 2 minutes to warm up. Pour the warm milk into the blender with the condensed milk and add the vanilla and rum. Turn the blender to medium speed and run for about 1 to 2 minutes. While the blender is running, add 4 whole eggs and 2 yolks, one by one, until fully incorporated. Pour the custard mixture into the baking dish previously coated with caramel. Take your baking dish and cover the top with aluminum foil, nice and tight. Place it in a bigger heavy dish/pan for a water bath. Pour enough hot water into the larger pot to reach halfway up the sides of the baking dish. Tip: If you squeeze the previous peeled lime into the water, the pot you are using will not turn black during the baking process. Slowly place on the middle rack of your hot oven and bake at 350 degrees for about 55 minutes, covered. Then remove the foil cover and bake for another 20 minutes. The custard is ready when you touch it and it feels firm but still wiggles in the middle. Carefully take the custard out of the oven, since the water inside the large pot will be very hot. Set aside to cool off, about 30 minutes. Once it is cold, place it in the fridge for at least 3 to 4 hours or leave overnight before serving. When ready to serve and unmold, dip the quesillo in a saucepan of hot water, or in a sink full of hot water a few seconds. It should loosen up. If not, run a knife around the edge to help it a bit. Invert the quesillo onto a large serving plate and pour the caramel on top.
