Heat oven to 400ºF. Place a piece of foil on the bottom rack of the oven to catch any drippings.
Place the flour, baking powder, salt, and baking soda into the bowl of a food processor and process for 3 to 4 pulses. Pour the mixture into a large mixing bowl and, using your hands, work the butter into the flour mixture until about half of the fat disappears and the rest is left in pea-size pieces. Make a well in the center of the mixture and add the buttermilk, stirring with a rubber spatula or wooden spoon just until it comes together. Turn the mixture out onto a piece of parchment or waxed paper that has been lightly dusted with flour, shape into a ball, wrap, and store in the refrigerator while you prepare the filling.
Combine the blackberries, sugar, water, and ginger in a large mixing bowl. Pour the mixture into a 10-inch cast-iron skillet and place over medium heat. Bring this to a simmer, decrease the heat to medium-low, and continue to cook, stirring occasionally, until the liquid is thick enough to coat the back of a spoon, about 15 minutes. Retrieve the dough from the refrigerator and gently drop it on the fruit mixture using a 1-ounce disher or large spoon, evenly distributing it over the top. Bake until the top is just starting to brown, 15 to 20 minutes. Remove from the oven and allow the grunt to cool for 15 to 30 minutes before serving.