Preheat oven to 350°F (175°C). Grease an 8×8-inch square pan and line with parchment paper, leaving an overhang on two sides.
Whisk flour, cocoa powder, baking soda, baking powder, and salt in a large bowl until fully combined.
Whisk melted butter, granulated sugar, and brown sugar in a second bowl. Add eggs one at a time, whisking after each. Stir in vanilla, mashed bananas, and buttermilk.
Pour wet ingredients over dry ingredients. Fold gently with a spatula until just combined and no white streaks remain. Fold in 1/2 cup chocolate chips. Do not overmix.
Pour batter into prepared pan and smooth the surface. Scatter remaining 1/4 cup chocolate chips on top. Bake for 32–35 minutes, until the edges are set and a toothpick inserted in the center comes out with moist crumbs, not wet batter. Cool in the pan for 15 minutes.
Make the frosting: In a small saucepan over medium-low heat, whisk together butter, cocoa powder, and milk until the butter is melted and the mixture is smooth and steaming. Do not boil. Remove from heat. Stir in sifted powdered sugar and vanilla until glossy and smooth.
Pour warm frosting over the slightly cooled cake. Spread gently to the edges. Scatter fleur de sel if using. Let set for 30 minutes at room temperature before slicing into squares with a warm, dry knife.