For the French toast: Put the babka slices on a rack set over a baking sheet and let dry for a few hours.
Preheat the oven to 200 degrees F; put a baking sheet with a rack inside the oven.
Whisk the eggs, egg yolks, sugar, vanilla, cinnamon and salt in a large bowl until smooth. Whisk in the half-and-half. Transfer the custard to a baking dish. Place a few slices of the babka in the custard and let sit, turning once, until completely soaked through, 5 minutes.
Heat 2 tablespoons of the butter and 2 tablespoons of the oil in a large nonstick saute pan over medium heat until the butter foams. In batches (so you don't crowd the pan), cook the bread until golden brown on both sides and just cooked through, about 2 minutes per side. Transfer the cooked French toast to the baking sheet to keep warm.
For the caramelized bananas: Melt the butter in a medium nonstick pan over medium heat. Add the sugar and cook until it just turns light golden brown, about 3 minutes. Place the sliced bananas in an even layer in the pan and cook until the bottoms become caramelized, about 2 minutes. Flip and cook another minute. Add the maple syrup and cook until just heated through, about 2 minutes.
Serve the French toast topped with caramelized banana and dusted with confectioners' sugar.