Preheat your oven to 350°F (175°C). Butter a 9x13 inch baking dish and set aside.
In a mixing bowl, combine the blackberries, sliced peaches, 1/2 cup of sugar, lemon juice, vanilla extract, and ground cinnamon. If you want a thicker filling, fold in the cornstarch gently to combine. Set the fruit mixture aside to marinate while preparing the batter.
In a separate bowl, mix together the flour, remaining 1/2 cup sugar, baking powder, and salt. Whisk to combine.
Pour in the milk and melted butter into the dry ingredients. Stir until just combined; it's okay if there are a few lumps.
Pour the batter evenly over the prepared baking dish, but do not mix or stir it with the fruit filling.
Spoon the blackberry-peach mixture over the batter, distributing it evenly without disturbing the batter underneath.
Bake for 35-45 minutes, or until the top is golden brown and the fruit is bubbling. You can check doneness by inserting a toothpick into the cobbler's batter; it should come out clean.
Remove the cobbler from the oven and allow it to cool for about 10-15 minutes before serving.
Notes
Serve warm with a scoop of vanilla ice cream on top, and sprinkle a pinch of cinnamon or a few fresh berries around the edge of the plate for an elegant touch.