Caramel: In a heavy-bottomed pan, bring the sugar and water to a boil over medium heat Stir with a wooden spoon, until a pale brown color develops. Remove from the heat, and let rest for 2 to 3 minutes, then pour into the cake pan, evenly coating the entire bottom. The caramel is extremely hot, so be careful when pouring it into the cake pan or molds.
In a saucepan over medium-high heat, add the condensed milk, the milk, the lime juice and zest, orange liqueur, and vanilla. Simmer for 12 to 15 minutes, stirring throughout to combine flavors.
In a mixing bowl, whisk the eggs and yolk together, then slowly temper into the milk mixture, constantly stirring the eggs into the warmed milk. Slowly incorporating the eggs will keep the eggs from scrambling. Pour milk mixture over the caramel in cake pan. Put the cake pan into a larger shallow vessel filled halfway up with water. Put the dish into the oven and bake for 40 to 50 minutes.
Remove the flan from oven and let cool. Refrigerate the flan overnight. To remove the flan from the pan, put the pan into a shallow container with about 1-inch of warm water. Let sit for about 10 minutes, which will soften the caramel and make it easier to remove.