Preheat oven to 350 degrees and prepare a 9 x 13 pan by lining it with aluminum foil and then spraying it with cooking spray.
In a large mixing bowl, combine the cake mix, butter, egg, vanilla and mix either with a mixer, a spoon or with your hands, kneading it together. The batter will be extremely thick and will be difficult to combine, but keep stirring or use your fingers and knead it together, adding the water if necessary. Press the dough into the prepared pan. It will be tacky, sticky and a bit tricky to work with, but keep pressing, spreading and finessing it. Sprinkle the white and dark chocolate chips evenly over the top.
In a mixing bowl, combine the sweetened condensed milk, cream cheese, 1/2 t. vanilla and stir until well combined, and pour mixture evenly over dough, using a spatula to gently spread and even it if necessary.
Bake for 24-27 minutes, or until edge of bars begin to barely brown and pull away slightly from the sides of the pan, or until the middle has barely set, taking care not to overbake. Note that the middle will not seem very well set and will be jiggly upon removing from oven and that is okay, you want to keep these gooey. They will continue to set up more after they have cooled. Allow bars to cool well before slicing and serving. Store in an airtight container in the refrigerator for up to a week or freeze for up to three months.