Preheat the oven to 350 degrees F. Butter
a large deep baking dish (about 9-by-13 inches).
Whisk together the eggs, cream, milk and vanilla in a large bowl. Whisk in the granulated sugar and brown sugar.
Cut the doughnuts into halves or quarters depending on the size of the doughnuts and the desired texture of your pudding. Toss the doughnuts in the egg mixture and allow them to sit until the majority of the mixture is soaked up, 15 to 20 minutes. Gently mix with a spatula or your hands if necessary.
Transfer the mixture to the prepared baking dish. Cover the dish with plastic wrap and aluminum foil and carefully place in a water bath. Bake for about 1 hour.