Heat oven to 350°F and grease a 12-cup muffin pan very liberally with butter.
Combine flour, sugar, baking powder, baking soda, and salt in a large bowl with a whisk. Make a well in the center of the flour mixture and add butter, mashed banana, eggs and vanilla. Whisk together wet ingredients in the well, then mix in flour just until wet (do not overmix). Spoon batter into muffin cups.
Bake until a toothpick inserted in the center of a cupcake comes out clean, about 25 to 30 minutes. Remove from pan and let cool completely before continuing.
For the Frosting:
Combine all ingredients in the bowl of a stand mixer and beat until smooth and fluffy, about 4 to 5 minutes.
Once cooled slice each cupcake in half horizontally. Slice off the top of the cupcake if very domed to create a flat surface, then place the bottom half over the top and using a sharp pairing knife trim the sides so they are even all the way around.
Spread 1 tablespoon of frosting on one side and top with the other half cupcake.