Preheat the oven to 375F. Peel, remove pits and slice peaches, add to the mixing bowl. Rinse blackberries and place in bowl. Sprinkle brown sugar, sugar, lemon juice, cornstarch and cinnamon over fruit and stir. Let sit for 5 minutes and stir again. Pour fruit into a 2.5-3 quart baking dish.
For the Topping:
In a medium mixing bowl, whisk flour, sugar, brown sugar, baking powder and salt together. Using a pastry blender, mix butter into flour mixture until mixture becomes crumbly.
Pour boiling water into flour mixture and stir until combined. Spoon mixture over peaches and blackberries, spread gently making sure to cover the fruit completely.
For additional Topping, optional:
In a small bowl, mix cinnamon and sugar together and sprinkle on top of the cobbler.
Bake uncovered for 45-55 minutes or until topping is cooked thoroughly and has started to brown. Let sit for 10 minutes. Serve with ice cream.