Preheat the oven to 350°F (175°C). Place a 9-inch round cake pan nearby.
Add the sugar to a small saucepan over medium heat. Allow it to melt, gently swirling the pan occasionally, until it becomes deep amber.
Immediately pour the caramel into the cake pan. Carefully tilt the pan to coat the bottom, then set aside to harden.
Add the eggs, sweetened condensed milk, evaporated milk, and vanilla to a blender. Blend for 20 to 30 seconds, just until smooth.
Pour the custard mixture over the caramel in the cake pan.
Place the cake pan inside a larger baking dish. Pour hot water into the larger dish until it reaches halfway up the sides of the cake pan.
Cover the cake pan loosely with foil. Bake for 50 to 60 minutes, until the edges are set and the center has a light wobble.
Remove the cake pan from the water bath. Cool for 1 hour, then cover and refrigerate for at least 4 hours or overnight.
Run a thin knife around the edge of the pan. Invert onto a rimmed serving platter and let the caramel sauce flow over the custard.