Preheat the oven to 375°F (190°C). Line a muffin tin with cupcake liners and set it aside.
In a medium bowl, mash the bananas with a fork until smooth. Set aside.
In a large bowl, whisk together the oats, 1 cup of flour, brown sugar, white sugar, baking soda, baking powder, cinnamon, and salt.
In another bowl, beat the eggs using a whisk until just mixed. Add the oil, milk, and vanilla extract, and stir to combine.
Add the mashed bananas to the wet ingredients and mix until smooth.
Gradually add the dry ingredients into the wet mixture, stirring until just combined.
Meanwhile, toss the blackberries with the remaining 2 tablespoons of flour to prevent them from sinking in the batter.
Gently fold in the blackberries, being careful not to overmix.
Divide the batter evenly into the muffin cups, filling each almost to the top.
Place the muffin tray in the oven and bake for 20–22 minutes, or until a toothpick inserted in the center comes out clean.
Allow the muffins to cool in the pan for 5 minutes, then transfer them to a wire rack to cool completely.
Enjoy warm or store in an airtight container for up to 3 days.