Preheat oven to 350F. Butter a 13x9 pan and set aside.
Combine cinnamon and sugar. Generously brush both sides of the bread slices with butter, then sprinkle with cinnamon-sugar mixture, reserving 2 tablespoons for topping. Toast in the oven for 8 to 10 minutes, or until golden brown, then allow to cool.
In the bowl of a stand mixer, combine the whole eggs, the egg yolks, the sugar, and the vanilla. Using the whisk attachment, whisk until combined. Reduce the speed to low, then add the cream and milk. Remove from mixer, and stir in the salt and mashed bananas.
Cut each slice of bread into quarters, and add to the bowl of custard. Stir until the bread is thoroughly coated, then pour into prepared baking pan.
Cover with plastic wrap and refrigerate overnight (or at least 4 hours).
Preheat oven to 325F. Remove plastic wrap from pan, dot with 2 tablespoons of butter, then sprinkle with reserved cinnamon-sugar mixture. Cover with aluminum foil - cut small holes to vent. Bake for 20 minutes, then remove foil, and bake for additional 20-25 minutes, or until a knife inserted in the middle comes out clean.
Caramel: In a saucepan, whisk together sugar, corn syrup, and water and bring to a boil. Cook over medium-high heat until a medium-dark amber color, swirling occasionally. Be very careful - caramel can go from "not quite dark enough" to "burnt" in a matter of seconds - don't leave the pan unattended! Take off heat and add the cream, butter, and sour cream. Stir together until smooth.
Serve warm, with bananas, caramel, and toasted walnuts on top.