Preheat the oven to 325 degrees. Lightly grease and 8-inch square cake pan.
Place the milk and 1/2 cup of the cream in a saucepan and bring to a boil over medium-high heat. Be careful not to let it boil over. Place the eggs, egg yolks, 1/2 cup of the sugar, salt, and vanilla in a bowl and gradually add the milk mixture, whisking all the while. Add the melted chocolate and mix until it is fully incorporated.
Place the bread cubes in the prepared pan and pour the custard over it, pressing down to dunk them. Let rest at least 15 minutes and up to an hour for the bread to absorb the custard.
Pour the remaining 1/2 cup cream over the top and sprinkle with the remaining 2 tablespoons sugar and transfer to the oven. Bake until the pudding is firm and a knife inserted comes out clean, about 1 hour.
Serve warm or at room temperature with caramel sauce, a dollop of sweetened whip cream or ice cream, and sliced bananas.