In a bowl, whisk together eggs, sugar, banana and oil until smooth.
Sift in flour, baking soda, baking powder and salt. Stir until just combined.
Pour the batter into the prepared loaf pan, and bake for 50 to 60 minutes.
Remove from oven and place on rack to cool.
To prepare buttercream, beat peanut butter and butter for a few minutes in a standing mixer with the paddle attachment on medium speed. Add powdered sugar and turn the mixer on low, beat until the sugar incorporates with the butter and becomes light and fluffy, about 3 minutes.