Stir together 2 tablespoons sugar and 1/4 teaspoon cinnamon in a small bowl; set aside. Whisk together flour, 2 tablespoons sugar, the baking powder, a pinch of salt, and the ginger in a medium bowl. Stir together milk and butter in a small bowl. Stir milk-butter mixture into the flour mixture. Set batter aside.
Gently fold together the raspberries, blackberries, lemon juice, remaining 3/4 cup sugar, a pinch of salt, the remaining pinch of cinnamon, and 2 tablespoons water in a large bowl. Transfer the berry mixture to a large straight-sided skillet. Cover, and bring to a boil over medium-high heat, stirring occasionally.
Drop 8 large dollops of batter on top of berry mixture using 2 spoons, spacing them evenly. Sprinkle dumplings with the cinnamon-sugar mixture. Cover; reduce heat to medium. Cook until the dumplings are cooked through and juices are bubbling, about 15 minutes. Serve warm, drizzled with cream.