Preheat your oven to 350°F (175°C). Place a 9-inch round cake pan or flan mold nearby and have a kettle of hot water ready for the water bath.
Make the caramel by heating 1 cup sugar in a heavy-bottomed saucepan over medium heat. Do not stir—just swirl the pan occasionally until the sugar melts completely and turns deep amber, about 8-10 minutes. Immediately pour into your flan pan, tilting to coat the bottom evenly. Work quickly as it hardens fast.
In a large mixing bowl, whisk the eggs until well beaten and slightly frothy. Add the sweetened condensed milk, evaporated milk, whole milk, and vanilla extract. Whisk until completely smooth and uniform in color.
Pour the custard mixture through a fine-mesh strainer directly into the caramel-coated pan to remove any egg solids. This ensures a silky-smooth texture.
Place the flan pan inside a larger roasting pan. Carefully pour hot water into the roasting pan until it reaches halfway up the sides of the flan pan, creating a water bath.
Bake for 55-65 minutes, until the custard is set around the edges but still jiggles slightly in the center when gently shaken. The top should look matte, not shiny.
Remove from the water bath and let cool to room temperature on a wire rack, about 1 hour. Cover with plastic wrap and refrigerate for at least 8 hours or overnight.
To unmold, run a thin knife around the edges of the pan. Place a rimmed serving plate upside down on top of the pan, then quickly flip both together. Let sit for 1 minute, then slowly lift off the pan, allowing all the caramel to drip onto the custard. Slice and serve chilled.