MAKE THE FILLING: Preheat the oven to 400 degrees. In a bowl, mix the peaches, tapioca, sugar, salt, and lemon juice. Transfer the filling to a 9-by-13-inch baking dish.

MAKE THE TOPPING: Using a spice grinder or a mortar and pestle, grind the lavender to a powder. In a bowl, combine the lavender, flour, oats, granulated sugar, baking powder, salt, and lemon zest. Cut in the butter using a pastry blender or two knives, until the mixture resembles coarse meal. Add the 1/2 cup of cream and the buttermilk; stir until the dough is just moistened.

Using two spoons, form 3-tablespoon mounds of the topping and arrange them evenly over the peaches. Brush with cream and sprinkle with the turbinado sugar. Bake the cobbler for 50 minutes, until the topping is golden brown and the fruit is bubbling. Let cool slightly before serving.