Preheat the oven to 350°F.
In the bowl of an electric mixer with a whisk or paddle attachment, combine the flour, sugar, milk, and vanilla. Beat at low speed for 1 minute to combine. Add the eggs and sour cream, increase the speed to medium-low and beat until they are combined. Add the goat cheeses and beat until they are combined, about 2 minutes.
In a 2-quart casserole dish or baking pan, or small ramekins, sprinkle the white chocolate evenly over the bottom, then pour in the beaten mixture. Place this dish (or pan or ramekins) in a larger pan and fill the larger pan with enough water to come halfway up the sides of the inner dish (or pan or ramekins). Bake until the surface appears firm and a knife inserted into the custard comes our clean, about 50 minutes.
When the custard is done, remove from the oven and invert onto a serving plate. If you baked your custard in a pan, cut it into squares. Serve with the Berry Sauce (recipe below).