In a saucepan, whisk the cornstarch with 1/4 cup of cold water until thoroughly combined and smooth. Stir in the blackberries, white sugar, and 1/2 cup brown sugar, and bring to a boil over medium heat. Stir until the filling thickens, about 2 minutes, then remove from heat and stir in vanilla extract. Pour into a 9x13-inch baking dish; set aside.
In a bowl, mix the cornflakes crumbs, flour, rolled oats, 1/3 cup brown sugar, pumpkin pie spice, and salt. Cut in 1/4 cup of butter with a pastry cutter until the mixture resembles coarse crumbs; stir in water, 1 tablespoon at a time, just until the crust begins to hold together. Sprinkle the crust over the berries, and drizzle with melted butter.
Bake the cobbler in the preheated oven until the crust begins to brown, about 10 minutes. Turn the oven's broiler on, and broil the cobbler until the crust is golden brown, about 2 more minutes.