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Preheat the oven to 350°F (175°C) and line an 8x8 or 9x9-inch square pan with parchment paper.
Brown the butter and let it cool slightly.
In a large bowl, whisk together the mashed bananas, browned butter, brown sugar, granulated sugar, eggs, Greek yogurt, and vanilla until smooth.
Fold in the dry ingredients until just combined. Do not overmix.
In a small bowl, combine the brown sugar, softened butter, flour, cinnamon, and salt for the cinnamon swirl.
Spread half of the batter into the prepared pan.
Dollop the cinnamon swirl over the batter, then gently spread the remaining batter on top.
Bake for 40–50 minutes, checking around 35 minutes, or until a toothpick inserted into the center comes out with a few moist crumbs.
Allow the cake to cool completely.
Spread the cream cheese frosting over the cooled cake.
Spoon the cinnamon swirl topping over the frosting and gently swirl it with a toothpick or butter knife.