Preheat the oven to 350 degrees (180 degrees Celsius). Generously grease a 9 x 13 inch (23 x 33 cm) baking dish with about 2 tablespoons of the butter.
If using the ripe bananas, mash them in a large bowl until no lumps remain. Add the milk, chia seeds, cornstarch, vanilla, cinnamon, and salt and whisk vigorously. Refrigerate for 10 to 12 minutes to allow chia seeds to jell. Remove from the fridge and stir briefly to break up any chia seed clusters. Set aside.
In a nonstick skillet over medium heat, heat the coconut oil. Cook the banana halves, flipping once, until golden brown and caramelized, about 2 minutes on each side. Using an offset spatula (or a fork, but be gentle), arrange the banana slices in a single layer on the bottom of the prepared baking dish.
In the same skillet you used for the bananas, add the brown sugar, 2 tablespoons maple syrup, and ¼ cup (60 ml) of water. Cook over medium heat, stirring, until the syrup is aromatic and the sugar is dissolved, 1 to 2 minutes. Pour the caramel over the fried bananas.
Working with one piece at a time, dip the sliced bread into the milk mixture, until the milk is just absorbed. Build layers on top of the bananas and caramel in the dish, overlapping the soaked bread slices so there aren’t any gaps. When you run out of bread, pour any leftover milk mixture on top. Press down on the bread to make sure all the slices are evenly coated and saturated. Break the remaining 2 tablespoons butter into small pieces and sprinkle over the top.
Bake for 35 minutes, or until golden. Let cool for about 15 minutes to give the caramel time to set, then run a large knife around the edge of the bread to loosen it from the dish. Place a cutting board or large platter on top of the baking dish. Use a kitchen towel or oven gloves to hold the two together securely. Take a deep breath and flip the cutting board and dish over in one swift move. Voilà!
Serve warm with coconut or vanilla ice cream (if using) and additional maple syrup. Leftovers can be stored, covered, in the refrigerator for up to 2 days. Warm in a low oven before serving.