In a medium saucepan, combine the blackberries with the sugar, 3 cups of the water and lemon juice. Cook until the berries are soft and a heavy syrup has formed, about 15 minutes.
Remove from the heat and strain off the syrup from the berries. Set the berries aside with 1/2 cup of the syrup and return the remaining syrup to the saucepan.
Add the remaining 1 cup water, the vanilla, and the butter to the syrup in the saucepan. Bring to a boil over medium-high heat.
For the dumpling dough:
In a large bowl, sift together the flour, sugar, baking powder, salt, and baking soda.
Using a pastry blender, 2 knives, or your fingertips, cut in the 6 tablespoons butter until the mixture resembles coarse crumbs.
In a small bowl, dissolve the yeast in the warm water. Add the yeast mixture and the buttermilk to the flour mixture, stirring only until just incorporated. Do not overwork the dough.
Drop the biscuit dough, by the tablespoonful, in batches, into boiling syrup and cook until the dumplings float to the surface and they are cooked through, about 5 minutes.
Remove the dumplings from the syrup and place in small bowls. Ladle some of the hot syrup and cooked berries over the top of the dumplings. Serve garnished with a dollop of whipped cream or a small scoop of ice cream.