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.