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.

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 in 30-second intervals, stirring after each interval, for 1 to 3 minutes.

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, approximately 5 to 6 hours.