Using the extra butter, grease an 8-inch square oven-safe dish, and set it aside.
Microwave about 3 tablespoons of the remaining butter in a small bowl or measuring cup until melted.
Mix the flour, 3 tablespoons of the granulated sugar, the baking powder, the baking soda, and 1/2 teaspoon of the salt into a large mixing bowl.
Mix together the buttermilk, 2 of the eggs, and the melted butter in a small bowl.
Add the wet ingredients into the dry ingredients, stirring the mixture until it has just combined and being careful not to overmix.
Heat a large non-stick skillet over medium heat.
Melt the remaining 1 tablespoon of the butter in the skillet.
In intervals, pour about 1/4 cup of the batter onto the skillet, cooking until the bottom is a golden-brown, about 2-3 minutes.
Flip and cook the reverse side of the pancake until it is golden-brown, about 2 minutes. Reduce the heat if the pancake browns too quickly.
Transfer the cooked pancakes to a plate and repeat the process until all of the batter is used, adding more of the extra butter to the skillet as needed.
In a large mixing bowl, mix the half-and-half, vanilla, the remaining eggs, the remaining granulated sugar, and the remaining salt together.
Put half the pancakes into the prepared baking dish.
Sprinkle 1/2 of the sliced bananas and 1/2 of the chocolate chips over the pancakes.
Pour 1/2 of the cream mixture evenly over the pancakes.
Repeat with the remaining pancakes, the remaining bananas, the remainig chocolate chips, and the remaining cream mixture.
Cover the dish with plastic wrap, and refrigerate for at least 8 hours and up to overnight to allow the pancakes to soak in the custard.
Preheat the oven to 350 degrees F.
Take off the plastic wrap, and cover the dish loosely with aluminum foil and bake for 30 minutes.
Uncover the dish and continue to bake until golden-brown, the custard is set and the center feels firm but springy, about 30 minutes.
Let the casserole cool for 15 minutes.
Dust with the confectioners' sugar and serve.