Preheat the oven to 375 degrees F.
Put the brioche cubes in a large bowl.
In a saucepan, add the cream, 1 cup of brown sugar and pinch of salt.
Bring to a simmer over medium heat. Whisk the eggs and yolks together in a large bowl.
In a slow stream, pour the hot cream mixture into whisked eggs to temper.
Strain through a fine mesh sieve over the brioche cubes (If you are very careful and temper correctly, you can eliminate this step)
Let this sit to allow the bread to absorb some of the liquid, about 10 minutes.
In a bowl, mash the bananas with 1/4 cup of the brown sugar.
Fold the mashed bananas into the bread cubes and transfer the mixture to an 8 by 11-inch baking dish. Sprinkle with the remaining 2 tablespoons of brown sugar and cover with foil. Bake until mostly set, about 35 to 40 minutes.
Remove the foil and bake another 15 minutes. Remove from oven when top is golden brown.
Eat and enjoy!