Combine 6 oz blackberries and 2 tbsp sugar in a small saucepan. Cook over medium heat, mashing occasionally, until berries break down and juices thicken slightly, about 8 minutes. Press through a fine-mesh sieve into a bowl, scraping with a spatula to extract all liquid. Discard seeds. You need 3-4 tbsp puree for the buttercream; reserve any extra for drizzling. Let cool completely.

Preheat oven to 350F. Line a 12-cup muffin tin with paper liners. In a large bowl, whisk together flour, cocoa powder, sugar, baking soda, salt, and crushed lavender until no lumps remain.

Add buttermilk, oil, egg, and vanilla to the dry ingredients. Whisk until just combined. The batter will be thick. Pour in hot coffee and whisk until smooth and glossy, about 30 seconds. Fold in halved blackberries gently so they don't break apart.

Divide batter among liners, filling each about two-thirds full. Bake until tops spring back when lightly pressed and a toothpick inserted in the center comes out with a few moist crumbs, 20-22 minutes. Cool in pan 5 minutes, then transfer to a wire rack to cool completely.

Beat butter on medium-high speed until pale and fluffy, about 3 minutes. Reduce speed to low and add powdered sugar 1 cup at a time, mixing until incorporated. Add 3 tbsp cooled blackberry puree, heavy cream, and salt. Beat on medium-high until light and spreadable, about 2 minutes. If too thick, add more cream 1 tsp at a time; if too thin, add more powdered sugar.

Spread or pipe buttercream onto cooled cupcakes. Top with fresh blackberries and a tiny lavender sprig if desired. Serve at room temperature.