Heat the oven to 375 degrees.
Line a 12-cup muffin pan with paper liners.
Whisk the flour, baking powder, salt, and sugar in a large bowl.
In another bowl, whisk the eggs, oil, milk, sour cream, and vanilla.
Pour the wet mixture into the dry mixture.
Stir just until the flour streaks mostly disappear.
Toss the blackberries with 1 tablespoon flour.
Fold the berries into the batter with a gentle hand.
Divide the batter between 12 muffin cups.
Sprinkle coarse sugar on top, if using.
Bake for 18 to 22 minutes.
Cool in the pan for 5 minutes.
Move muffins to a rack before serving.