Preheat the oven to 425 °F (218 °C) and line 2 muffin pans with 16 liners.
Mix the flour, brown sugar, granulated sugar, baking powder, baking soda, and salt in a large mixing bowl until thoroughly incorporated.
1 3/4 cups (219 grams) flour, 1/2 cup (100 grams) light brown sugar, 1/2 cup (100 grams) granulated sugar, 2 1/2 teaspoons baking powder, 1/2 teaspoon baking soda, 1/2 teaspoon salt
Whisk the mashed banana, oil, eggs, vanilla, and sour cream together in a medium mixing bowl.
1 1/2 cups (370 grams) very ripe, spotted, mashed bananas (about 3.5 medium bananas), 1/4 cup (52 grams) olive oil, 2 large eggs, 2 teaspoons vanilla extract, 1/4 cup (60 grams) sour cream
Pour the wet mixture into the dry mixture, stirring just until the ingredients are combined.
Carefully fold the blackberries into the batter.
2 cups (272 grams) fresh or frozen blackberries
Divide the batter evenly among the prepared liners, filling each cup until almost full (I used 80 grams), trying to make sure there are some blackberries visible at the top of the muffins (at least with some of the muffins).
Evenly sprinkle the coarse sugar over the tops.
2-3 tablespoons coarse sugar
Bake the muffins for 10 minutes and then lower the temperature to 350 °F (175 °C) and continue baking for 4-7 additional minutes or until a toothpick inserted into the middle of one of the center muffins comes out clean.
Let the baked muffins sit in the pan for 5 minutes before moving them to a wire rack to finish cooling.
These muffins are best the day they’re made, while the coarse sugar topping is still crisp. For the first day, store them loosely covered at room temperature. This helps the sugar topping stay crisp longer. After that, transfer the muffins to an airtight container so they don’t dry out. The sugar topping will soften, but the muffins will stay soft and moist for up to 3 days at room temperature or up to 5 days in the refrigerator. For longer storage, freeze the muffins in an airtight container or freezer bag for up to 3 months. Thaw at room temperature or microwave briefly to warm through.