How to Make It
Preheat the oven to 350°. Butter an 8-by-11-inch baking dish and coat the dish with the turbinado sugar.
In a large saucepan, bring the cream, milk, 2/3 cup of the granulated sugar and the salt to a simmer over moderately high heat, then remove from the heat.
In a bowl, whisk the whole eggs, egg yolks and vanilla. Gradually whisk in the hot cream until blended. Strain the custard through a fine strainer into a large bowl.
In a small bowl, toss the blueberries and raspberries with 2 tablespoons of the granulated sugar. Using a fork, coarsely mash the berries. Let stand until juicy, about 5 minutes.
Mix the brioche into the custard. Fold in the mashed berries. Transfer the pudding to the prepared baking dish and sprinkle the top with the remaining 2 tablespoons of granulated sugar. Cover with foil and bake for 45 minutes, until set in the center. Remove the foil and bake in the top third of the oven for 20 minutes longer, until lightly golden.
Preheat the broiler. Broil the pudding for 1 minute, until golden brown. Transfer to a rack and let stand for 30 minutes, until cooled slightly. Cut the bread pudding into squares and serve with blueberries, raspberries and whipped cream.