Heat the oven to 325°F (165°C). Place a 9-inch round cake pan inside a deep roasting pan; put a kettle of water on to boil for the water bath.
Make the caramel: Combine the granulated sugar and 0.25 cup water in a medium saucepan over medium heat. Cook without stirring until the syrup turns a deep amber, 6–9 minutes, swirling the pan gently near the end for even color.
Immediately pour the caramel into the cake pan and tilt to coat the bottom evenly. Set aside to let the caramel harden.
In a large bowl, whisk the eggs and egg yolks just until blended, avoiding excess foam.
Whisk in the sweetened condensed milk, evaporated milk, vanilla extract, and fine salt until smooth.
Strain the custard through a fine-mesh sieve into a large measuring cup or pitcher to remove bubbles and chalazae.
Pour the custard into the caramel-lined pan. Cover the pan tightly with foil to promote even cooking.
Set the roasting pan with the filled cake pan on the middle oven rack. Carefully pour about 8 cups boiling water into the roasting pan to come halfway up the sides of the cake pan.
Bake until the edges are set and the center still jiggles like gelatin, 50–70 minutes; an instant-read thermometer inserted in the center should read 175–180°F (79–82°C).
Remove the cake pan from the water bath, discard foil, and let cool on a rack for 1 hour.
Refrigerate uncovered until thoroughly chilled and set, at least 4 hours or up to 24 hours.
To unmold, briefly set the bottom of the pan in hot water to loosen the caramel. Run a thin knife around the edge, invert a rimmed serving plate over the pan, and flip to release. Let the caramel sauce flow over the custard. Slice and serve cold.