Add blackberries, ½ C sugar and cornstarch to a 9x9 baking dish. Stir well to cover the berries. Place into preheated oven and cook for 20 minutes until the berries begin to bubble.
While the berries are cooking prepare the topping.
Combine coconut milk and vinegar in a mixing cup and set aside. In a large bowl, combine flour, baking powder, remaining ¼ C sugar and salt. Stir together. Add in the coconut oil and vanilla. Work together with a fork until it is small crumbles. Add in the sour milk and fold together with a spatula.
Scoop generous dollops of the topping on to the cooked blackberries. Sprinkle the top with the coarse sugar and cinnamon.
Bake an additional 20 minutes until top is golden brown.