In a wide, shallow bowl, whisk together the eggs, whole milk, heavy cream, mashed bananas, granulated sugar, cinnamon, vanilla extract, and salt until the mixture is well-combined.
Place a large non-stick skillet or griddle over medium-low heat and melt 1 tablespoon of butter, swirling to coat the surface.
Dip the bread slices into the banana custard, allowing each side to soak for approximately 20-30 seconds until saturated but not soggy.
Transfer the soaked bread to the skillet and cook for 3 to 4 minutes per side, or otherwise until golden brown and the custard center is set. Work in batches to avoid crowding the pan.
In a separate small skillet, melt the remaining 2 tablespoons of butter over medium heat.
Stir the brown sugar and maple syrup into the melted butter until the mixture is bubbling and the sugar is dissolved.
Add the sliced bananas to the syrup and sauté for 2 minutes, tossing gently until the fruit is softened and coated in a dark caramel glaze.
Serve the warm French toast immediately, topped with the caramelized bananas and a drizzle of the remaining maple-butter sauce from the pan.