Preheat the oven to 400°F.
Place the sliced ripe bananas in an 8-inch square baking dish.
Add the dark brown sugar and 1 tablespoon of chilled butter pieces.
Bake for 35 minutes, stirring once after approximately 17 minutes.
Remove the roasted banana mixture from the oven and allow it to cool slightly.
Reduce the oven temperature to 375°F.
In a large mixing bowl, combine the cake flour, baking soda, baking powder, and cinnamon.
Set the dry mixture aside.
In a separate medium bowl, combine the roasted banana mixture, buttermilk, and vanilla extract.
Stir until evenly combined, then set aside.
In the bowl of a stand mixer fitted with the paddle attachment, beat the softened butter and granulated sugar on medium speed until well blended.
Add the eggs.
Mix until fully incorporated.
Add the flour mixture to the butter and sugar mixture alternately with the roasted banana mixture.
Begin and end with the flour mixture.
Mix after each addition only until combined.
Coat a 13x9-inch baking pan with baking spray.
Pour the batter into the prepared pan.
Spread the batter evenly.
Bake at 375°F for approximately 20 minutes, or until a wooden pick inserted into the center comes out clean.
Remove the pan from the oven.
Allow the bars to cool completely in the pan on a wire rack before frosting.
To prepare the frosting, melt 1/4 cup butter in a small saucepan over medium heat.
Cook for approximately 4 minutes, or until the butter is lightly browned and has a nutty aroma.
Remove from the heat and allow the browned butter to cool slightly.
In a medium mixing bowl, combine the browned butter, powdered sugar, softened cream cheese, and vanilla extract.
Beat with a mixer until smooth and creamy.
Spread the frosting evenly over the cooled banana bars.
Sprinkle the toasted chopped pecans over the frosting.
Allow the frosting to set slightly before slicing.
Cut into bars and serve.