How to Make It
Make the cake: Preheat oven to 350°F. Line an 8-inch square metal baking pan with parchment paper, allowing paper to overhang about 2 inches.
Melt butter in a medium skillet over medium. Continue cooking until butter turns light brown and smells incredibly nutty-fragrant, 4 to 5 minutes. Remove from heat; cool slightly.
Whisk together flour, baking soda, and salt in a medium bowl. Make a well in center of mixture; crack in egg, and whisk to beat. Add banana, brown sugar, buttermilk, and vanilla; whisk until well blended. Stir in browned butter. Fold in chocolate chips. Spoon mixture into prepared baking pan.
Bake in preheated oven until cake springs back when lightly touched in the center and a wooden pick comes out clean, 30 to 33 minutes. Let cool in pan on a wire rack 5 minutes. Remove cake from pan, using parchment as handles, and let cool on rack to room temperature, about 1 hour.
Make the frosting: Combine peanut butter, butter, vanilla, and salt in a large bowl; beat with a handheld mixer on medium speed until fluffy, about 1 minute. Add powdered sugar; beat on low speed until just combined. Increase speed to medium, and beat until smooth. Spread frosting over cake. Garnish with chocolate chips, if desired.Store at room temperature for up to 3 days or in the refrigerator up to 1 week.