Preheat the oven to 350° F, and lightly and oil 8 (2x4-inch) mini loaf pans.
In a large bowl, combine the flour, baking soda, and salt; set aside. Place the bananas in another large bowl and mash them well with a whisk or a potato masher or a fork. Add the brown and granulated sugars and the melted butter and beat with an electric mixer or a whisk until well blended (there will be little bits of banana that don't get perfectly smooth). Beat in the eggs one at a time, and then beat in the vanilla. Add the dry ingredients, switch to a spoon, and stir just until the flour is incorporated.
Spoon the mixture evenly into the tins (they will be fairly full), and bake until a wooden skewer inserted into the middle of the mini loaves comes out clean, about 25 to 28 minutes. Let cool for a few minutes in the tins on a wire rack, then turn the little loaves out onto the rack to finish cooling. Serve warm or at room temperature.