Preheat the oven to 350 degrees. Line two 12 cup muffin pans with 20 liners.
In a large mixing bowl, whisk together the melted butter, brown sugar, and white sugar until smooth, creamy, and no brown sugar lumps remain.
Whisk in the sour cream, then add the eggs, one at a time, whisking well after each addition. Stir in the vanilla.
Add the bananas on top of the mixture, and mash them with a potato smasher or fork, then stir them into the batter.
Sprinkle 1 ½ cups flour, baking soda, baking powder, salt, and cinnamon on top of the wet ingredients and give them a little whisk. Then use a spatula to gently fold everything together until no white remains.
Toss the blackberries with the remaining tablespoon of flour, then pour them into the batter. Gently stir the berries until well combined.
Scoop the batter into the prepared liners. Sprinkle each muffin top with turbinado sugar, if desired. Bake for 20-25 minutes, or until the tops of the muffins are set and spring back when touched. Cool for a few minutes in the pans, then remove the muffins to a cooling rack to cool completely.