Wash the blackberries gently and pat them dry. In a medium bowl, toss the berries with 1/2 cup sugar, cornstarch, and lemon juice. Let sit for 10 minutes.
In a separate bowl, whisk together flour, remaining 1 tablespoon sugar, baking powder, and salt. Cut in the cold butter using your fingers or a fork until the mixture resembles coarse crumbs. Stir in the milk just until a soft dough forms. Do not overmix.
Lightly grease your 10-inch Dutch oven or camp skillet. Pour the berry mixture into the pot. Drop spoonfuls of the dough over the berries, spacing them evenly. The dough will spread as it cooks.
Cover the Dutch oven with its lid. Place it on a camp stove over low to medium-low heat. Cook for 30-35 minutes, rotating the pot every 10 minutes to prevent burning. The cobbler is done when the topping is golden and the berries bubble thickly around the edges.
Remove from heat and let rest for 5 minutes. Serve warm, straight from the pot. A dollop of plain yogurt or a scoop of vanilla ice cream makes it even better — if you have a cooler.