Heat the oven to 220°C / 425°F and line a 12-cup muffin pan with parchment tulip liners or standard muffin liners. If using frozen blackberries, keep them frozen until the moment they're folded in.

Cut any very large blackberries in half, leaving smaller ones whole. Pat fresh berries dry gently, then toss them with the tablespoon of flour. Don't crush the berries while coating them; you want distinct pockets and streaks, not fully purple batter.

Whisk together the flour, cornmeal, baking powder, baking soda, and salt in a medium bowl, mixing thoroughly so the leavening distributes evenly through the cornmeal. Use medium-grind cornmeal rather than coarse polenta; polenta stays too hard and makes the crumb rough instead of tender.

Whisk together the melted butter, oil, and sugar in a large bowl, then whisk in the eggs until the mixture looks smooth and slightly thickened. Add the sour cream, buttermilk, and vanilla, whisking until fully combined and smooth. The butter should be melted but not hot; hot butter can split the mixture or cook the eggs in spots.

Add the dry ingredients to the wet ingredients and fold gently with a spatula until the flour is mostly incorporated, with a few streaks still visible. Add the blackberries and fold just until evenly distributed. The batter should be thick and scoopable; don't overmix once the flour goes in, since that tightens the crumb and bursts too many berries, turning the batter gray-purple.

Let the batter rest for 10 minutes at room temperature. This helps the cornmeal hydrate and softens the final crumb while also encouraging better muffin height. Don't skip this step if you want the coarse-but-tender texture this muffin is meant to have.

Divide the batter evenly among the 12 muffin cups, filling each about 85 to 90% full for tall domed tops. Press 1 to 2 extra blackberries lightly into the tops if desired, so some berry pockets stay visible after baking, then sprinkle lightly with coarse sugar.

Bake at 220°C / 425°F for 5 minutes, then, without opening the oven, reduce the temperature to 190°C / 375°F and continue baking for 13 to 17 minutes, until the tops are high and golden, the edges are lightly browned, the center springs back when pressed lightly, and a tester inserted into a non-fruit part of the muffin comes out clean or with a few moist crumbs. Don't test through a blackberry pocket or melted fruit streak, or the muffin may seem underbaked when it's actually done. The crumb should be moist and tender with visible cornmeal texture, not wet in the center and not dry or crumbly.

Let the muffins cool in the pan for 5 to 8 minutes, then transfer to a rack and cool another 10 to 15 minutes before serving. They're best slightly warm or at room temperature.