Preheat oven to 425ºF. Butter a 9 X 13 inch glass or ceramic baking dish (3-quart capacity).
In a large bowl, whisk together 1½ cups sugar and cornstarch, then add the blackberries and peaches and toss gently to combine. After starting to mix, add cinnamon, nutmeg and vanilla, continue to toss and combine well. Transfer to prepared baking dish and bake until just bubbling, about 10 - 15 minutes.
Meanwhile, make the dough by whisking together the flour, baking powder and salt in another large bowl. Add the butter and blend with your fingertips or pastry cutter until mixture resembles coarse meal. Add milk and stir with wooden spoon just until a dough forms.
Using a large ice cream scoop, drop dough onto hot fruit mixture in 12 mounds (about ⅓ cup each), any extra dough can be dropped evenly around the mounds. Sprinkle dough with remaining teaspoon of sugar. Bake cobbler in center of oven until top is golden, 25 - 35 minutes. Let cool on a wire rack for a bit, then serve warm with a scoop of vanilla ice cream.