Set a large bowl of ice water next to your stovetop. Place the sugar and 1/4 cup of water in a medium saucepan over medium-high heat. Using a pastry brush, stir the sugar occasionally, being careful not to get any grains stuck against the sides of the saucepan, until it has all dissolved. Swirl the saucepan every minute until the sugar becomes a golden reddish-brown, about 10 to 15 minutes. Shock the bottom of the saucepan by placing the bottom in the ice water to stop the cooking, and then equally divide the caramel into 10 to 12 (3 1/2-inch) ramekins and set aside.
Preheat your oven to 325 degrees F.
Using an electric mixer, beat the softened cream cheese with the eggs on low speed to combine. Increase the speed to medium-high and beat until completely incorporated. Add the condensed milk, evaporated milk, whole milk and vanilla extract and continue to beat together until everything is well incorporated, scraping down the sides of the bowl as necessary, about 2 to 3 minutes.
Ladle the custard over the caramel into the ramekins filling them up to 1/2-inch from the rim. Place a kitchen towel in a deep baking dish or roasting pan and place the ramekins in 2 rows on top of the towel. Place the baking dish in the oven and using a pitcher, pour enough hot water into the baking dish to reach halfway up the sides of the ramekins, taking care not to get water in the ramekins. Cover the baking dish tightly with aluminum foil and bake until the flan is just set, about 30 minutes. When you tap the edge of the ramekin the flan should still wobble in the center.
Carefully remove the baking dish from the oven, remove the foil and let the ramekins cool slightly. Using tongs remove the ramekins from the water and set on a towel to cool for at least 2 hours before serving. Once cooled completely, the flan can be refrigerated for up to 3 days before un-molding and serving. To serve, run a thin knife around the inside edge of the ramekin to loosen the flan, and then invert onto a plate.