Begin by preheating your oven to 350°F (175°C). Grease and line a 8x8” square baking pan with parchment paper, leaving some excess hanging over the sides for easy removal later. Set it aside.
In a saucepan over medium-high heat, melt the butter, stirring constantly. Continue cooking until the butter turns a rich amber color and releases a nutty aroma, about 5-8 minutes. Remove from the heat and allow it to cool slightly.
In a medium bowl, whisk together the flour, baking soda, salt, and cinnamon. Set this dry mixture aside.
In a large bowl, combine the slightly cooled brown butter with both brown and granulated sugars. Whisk until the mixture is smooth and well incorporated.
Next, add the eggs, yogurt, mashed bananas, and vanilla extract to the butter-sugar mixture. Whisk until everything is well combined.
Gradually add the dry ingredients to the wet mixture, stirring gently until just combined. Be careful not to overmix.
Pour the batter into your prepared baking pan and spread it evenly.
Bake at 350°F (175°C) for 45-55 minutes, until the cake turns a deep golden brown. Insert a toothpick into the center—if it comes out clean, the cake is done. If the top browns too quickly while the center is still undercooked, cover the pan loosely with foil for the final 15 minutes.
While the cake is cooling, make the frosting. In a medium bowl, beat together the softened cream cheese and butter until smooth. Add the vanilla extract and powdered sugar, and continue beating until the frosting is light, fluffy, and well combined. Set it aside for frosting the cooled cake.
Once the cake has completely cooled, spread the cream cheese frosting evenly over the top. Next, drizzle the caramel in dollops across the surface, and gently swirl it into the frosting using a spoon.
Slice into 16 squares and enjoy!