Preheat your oven to 400F (205C). Grease a muffin tin or add muffin liners.
In a large bowl, combine the melted butter, brown sugar, granulated sugar, vanilla extract, eggs, mashed bananas, and sour cream. Mix until fully combined.
Sift the all-purpose flour, baking powder, baking soda, and salt over the wet ingredients. Gently fold everything together to combine without overmixing.
Allow the batter to rest for about ten minutes.
Fill the muffin liners to the top, sprinkling a bit of brown sugar on top for added sweetness if desired.
Bake for 5 minutes at 400F (205C), then reduce the temperature to 350F (180C) and bake for an additional 12-15 minutes.
Check for doneness by inserting a toothpick; it should come out clean.
Let the muffins cool on a wire rack for about 5 minutes before serving.
Store any leftovers in an airtight container.