Make the bread pudding: Preheat the oven to 350 degrees F and butter a shallow 2-quart baking dish. Toss the croissants with the melted butter on a baking sheet. Arrange in a single layer and bake until toasted, about 10 minutes.
Meanwhile, whisk the eggs, egg yolks, milk, heavy cream, sugar, vanilla and salt in a large bowl. Arrange about half of the croissants in the prepared baking dish. Top with about two-thirds of the milk chocolate, then add the remaining croissants. Pour the egg mixture over the croissants, pressing gently so the croissants are submerged.
Cover the baking dish with foil and bake 30 minutes. Uncover, top with the remaining milk chocolate and bake until the bread pudding is puffed, golden brown and set, 15 to 20 more minutes. Let sit 15 minutes.
Meanwhile, make the sauce: Bring the heavy cream, butter, sugar, corn syrup and vanilla to a simmer in a medium saucepan over medium heat. Add the bittersweet chocolate and whisk until smooth. Serve the bread pudding with the chocolate sauce.