Preheat oven to 350 degrees. Grease an 8 or 9 inch springform pan with cooking spray and line with a parchment round, if desired.
In a mixing bowl or the bowl of an electric mixer, whisk together flour, sugar, baking powder, salt, and lime zest.
Add the softened butter, sour cream, milk, egg, and vanilla to the bowl and beat on low speed until moistened, then increase the speed and beat until a thick, batter forms. There may still be some small lumps of butter. That's okay.
Toss the blackberries with the 1 1/2 Tablespoons of flour, then very gently stir them into the batter. Scoop the batter evenly into the prepared pan. Note: I find it easiest to scoop small portions around the pan then join them together, since the batter is thick and hard to spread.
For the topping, combine the sugars, flour, and salt in a bowl. Use a pastry blender or your fingers to cut the butter into the flour mixture, working it in until the mixture resembles coarse crumbles. Sprinkle the crumble mixture evenly over the batter.
Bake for 40-45 minutes or until the cake is golden and lofted and a toothpick inserted in the center comes out clean.
While the cake cools, whisk together the powdered sugar, lime juice, and zest. Add a pinch of salt, to taste. Release the cake from the springform pan, drizzle with the glaze, and serve.