Preheat oven to 450° with rack in middle.
In a large bowl, blend together flour, fat and salt in a bowl with your fingertips or a pastry blender just until mixture resembles coarse meal with the occasional lentil-sized piece.
Drizzle water evenly over mixture. Gently stir with a fork until incorporated, then bring dough together with your hands. Divide dough in two parts, one a little bigger than the other, and flatten each to about 1 inch thick. Wrap in plastic and chill 15 to 30 minutes.
Roll the larger piece of dough on a floured surface with a lightly floured rolling pin into a 12-inch square, then tuck into an 8-inch square, 2-inch deep baking pan.
Roll smaller piece of dough into an 8-inch square so it will fit the top of the cobbler.
Stir together all bout about 1 tablespoon of the sugar with the cornstarch. Fill the crust in the pan with blackberries and sprinkle sugar mixture over top. Top with remaining dough. It doesn't matter if the crusts aren't joined; simmering filling can come up in cracks. Brush the top with milk or cream (which helps the pastry brown better) and sprinkle with reserved sugar. Put cobbler in oven and reduce temperature to 425°. Bake until crust is golden-brown and filling is bubbling, about 45 minutes. Cool at least 45 minutes before serving. Serve with ice cream or beaten unsweetened whipped cream.