Make the banana sauce: Melt the butter in a large stainless-steel skillet with a lid over medium heat. Add the brown sugar and whisk until it melts into the butter. Add the water and whisk until smooth. Cover and set aside while you prepare the French toast.
Prepare the French toast: Whisk the whole eggs and egg yolks in a large bowel to blend. Gradually whisk in the milk, cream and vanilla. Dip the bread slices into the egg mixture, turning to coat and soak until they are completely moistened with the custard.
Heat a heavy, large griddle pan over medium heat. Melt 1 tablespoon of butter on the griddle. Working in batches, transfer the soaked bread to the pan with a slotted spatula and cook until golden brown and cooked through, about 3 minutes per side. Add more butter as needed for each batch.
To serve, add the banana slices to the sauce and cook until the bananas are almost tender, tossing gently, about 3 minutes. Transfer the French toast to plates. Top each with the banana sauce and whipped cream, if desired.