In a large heatproof bowl, whisk the yolks and eggs. Slowly whisk in the sugar and salt until thoroughly combined. Pour the half-and-half into a medium saucepan. Heat over medium-high heat until steaming but not bubbling. Add the 2 cups chopped bittersweet chocolate to the half-and-half and whisk until the chocolate is completely melted. Slowly whisk the half-and-half mixture into the egg mixture until thoroughly combined. Strain the mixture through a fine sieve into a large Pyrex measuring cup or heatproof bowl. Add the vanilla extract.
Put the bread cubes in a 9x13-inch baking dish and pour the custard on top. Make sure the bread is as submerged in the custard as possible and let cool at room temperature for about an hour. Cover with plastic wrap and refrigerate for at least 5 and up to 24 hours.
Heat the oven to 325° F. Transfer the bread mixture to a large mixing bowl and gently fold in the bananas. Return the mixture to the baking dish. Cover the pudding loosely with foil and bake at 325°F for 70 minutes. Remove the foil and continue to bake until no liquid custard is visible when you poke a small hole in the center with a paring knife, 20 to 40 minutes more.
Let the pudding cool on a rack. Serve warm, at room temperature, or chilled, with a dollop of whipped cream, if desired.
Photo: Scott Phillips