Combine white and brown sugar and water in a saucepan over medium-high heat. Stir occasionally to dissolve the sugar completely, but stop stirring once the mixture comes to a boil to prevent the sugar from crystallizing.
Swirl the pan gently to redistribute the caramel as the sugar begins to darken. Cook to about 310 degrees F (160 degrees C), or until the sugar turns light golden brown. Remove from heat immediately, as sugar will continue to darken.
Pour the caramel in an 8-inch baking dish, coating the bottom and sides of the dish. Let caramel harden, about 5 minutes.
Preheat an oven to 325 degrees F (165 degrees C).
Melt the white chocolate in a microwave-safe glass or ceramic bowl in 30-second intervals, stirring after each melting, for 1 to 3 minutes (depending on your microwave). Do not overheat or chocolate will scorch.
Cream together the cream cheese, mascarpone cheese, evaporated milk, sweetened condensed milk, eggs, and melted white chocolate in a bowl until smooth.
Pour the cream cheese mixture over the hardened caramel.
Place baking dish in a large roasting pan. Fill the roasting pan with about 1-inch water.
Bake in the preheated oven until a knife inserted into the center comes out clean, about 1 hour.
Refrigerate until completely chilled, 5 to 6 hours. To serve, run a paring knife along the edges of the baking dish to loosen the flan. Cover the baking dish with a plate, and invert it to tip the flan out of the pan and onto the plate.