Toss peaches and blackberries with ginger, cinnamon, sugar, lemon juice, and cornstarch in a 2-qt. non-reactive baking dish and bake in middle of oven 10 minutes.
In a separate bowl, stir together flour, sugar, baking powder, and salt. Blend in butter with your fingertips or a pastry blender until mixture resembles coarse meal. Add the boiling water and stir until just combined.
Remove peaches and berries from oven and drop spoonfuls of topping over them. Sprinkle the 2 Tbs. of sugar over the cobbler topping. Bake in middle of oven until topping is golden, about 25 minutes. (Topping will spread as it bakes.)
Let cool for at least one hour. Serve with vanilla ice cream or whipped cream.