To make the sauce: Put the blackberries and water in a nonreactive saucepan.
Crush the berries gently with the back of a wooden spoon and bring to a low simmer.
Cook partially covered for 25 minutes, stirring occasionally.
Strain the berries through a fine mesh sieve, pressing gently on the solids until all the juice is extracted.
Discard the solids, and add the salt to the juice along with ¾ cup of the sugar.
Taste for sweetness.
The juice should be rather sweet.
Set aside to cool.
Preheat the oven to 425F.
To make the dumplings: Sift together the flour, baking powder, sugar and salt into a mixing bowl.
Using your fingers, quickly work the chilled butter into the flour mixture, until it is the texture of oatmeal.
Pour the half and half over and stir just until the dough comes together.
Turn the dough onto a lightly floured surface, and knead six to eight turns, with floured hands.
Pat or roll the dough into a disc ½ " thick.
With a biscuit or cookie cutter, stamp out 8 rounds 2 1/2" to 3" in diameter.
Arrange the cut dumplings in a buttered 2 quart baking dish and pour the cooled blackberry sauce over them.
Bake in preheated oven for 20 minutes.
Remove the baking dish from the oven and carefully baste the dumplings.
Then dot the tops with small pieces of butter and sprinkle the crushed sugar cubes over.
Return to the oven and continue baking for an additional 10-15 minutes until the dumplings are golden brown.
Serve the dumplings warm with some of the blackberry sauce spooned over and a large scoop of vanilla ice cream.