Preheat oven to 350 degrees. Butter a 9x13 baking pan.
Sift together the flour, baking powder, and salt.
Place butter and brown sugar in a stand mixer fixed with the paddle attachment. On medium speed cream the butter and sugar together, stopping to scrape the sides as needed, until the mixture is uniform and a pale light brown color, about 2 minutes.
Add the eggs one at a time beating them on medium speed until uniform.
Add half of the flour mixture, and on low speed, mix until just combined. Add all of the mashed banana, and on medium speed, mix until just combined, scraping the sides as neded. Repeat with the remaining flour mixture.
Transfer half of the batter to the baking pan and smooth the top with an offset spatula. Evenly top with the peanut butter and sliced bananas.
Evenly top with the remaining cake batter, and smooth the top with an offset spatula.
Bake for 26 to 30 minutes, until a toothpick placed in the center of the cake comes out clean (the top will be lightly browned and the edges will be a deep brown).
Place the cake on a cooling rack and allow to cool completely, about 1 hour 30 minutes.
When the cake is just about completely cooled, combine the butter and honey in a stand mixer fitted with the paddle attachment. Mix together on medium speed until uniform.
Add the powdered sugar and mix on low speed until uniform.
Dissolve the salt in the heavy cream and add to the sugar mixture.
Gradually turn the speed of the mixer to high, and mix the buttercream, scraping the sides as needed, until it is white and fluffy, about 3 minutes.