Place a rack on upper oven shelf. Preheat oven to 400°F. Butter a 9-by-13-inch baking dish. Line a large rimmed baking sheet with foil.
Mix flour, granola, baking powder and ginger with 1/2 cup sugar in a large bowl. Add butter and use fingertips, a pastry blender or two knives to work into a mixture that resembles coarse crumbs. Cover and refrigerate until ready to use. (Crumb mixture can be made up to 2 days in advance.)
Combine blackberries and nectarines in a large bowl. Add lemon juice, remaining 3/4 cup sugar and cornstarch; stir well and pour into baking dish. Sprinkle crumb mixture evenly over fruit. Place baking dish on lined baking sheet and bake until browned and crisp, 40 to 45 minutes. Let cool for at least 35 minutes before serving.