Preheat the oven to 400°F(204°C). Line a 12-cup muffin pan with paper liners.
In a medium mixing bowl, whisk together the flour, almond flour, oats, salt, sugar, and baking powder.
In a separate medium mixing bowl, mash the bananas with a fork until smooth. Add the eggs, buttermilk, vanilla, and avocado oil, and stir until well combined.
Add the dry ingredients to the wet mixture and gently fold together with a rubber spatula until just combined. It's okay if a few streaks of flour remain—be careful not to overmix.
Roughly chop blackberries and toss them with 1 tablespoon of flour, then fold into the muffin batter.
Divide the batter evenly among the paper liners (they will be almost full). If desired, sprinkle the tops with Turbinado sugar and rolled oats.
Bake for 10 minutes at 400°F/204°C. Drop the oven temperature down to 375°F/190°C and bake for another 15 minutes. Muffins are done when a cake tester comes out clean.
Cool in the pan for 10 minutes. Remove the muffins from the pan and cool completely.