Preheat your oven to 350°F (175°C). Generously grease a 9x13 inch baking dish with butter or cooking spray.
In a medium mixing bowl, gently combine the fresh blackberries and sliced peaches. Sprinkle ½ cup of the granulated sugar over the fruit, then add the cornstarch, vanilla extract, and lemon juice. Using a spatula, toss the mixture gently until the fruit is well-coated. Transfer the fruit filling to the prepared baking dish, spreading it out evenly.
In another large mixing bowl, whisk together the flour, baking powder, salt, the remaining ½ cup of sugar, and ground cinnamon until combined.
Create a well in the center of the dry ingredients and pour in the melted butter and milk. Stir gently with a wooden spoon or spatula just until the ingredients are mixed; do not overmix. Expect the batter to be slightly lumpy.
Using a spoon or spatula, dollop the batter evenly over the fruit mixture in the baking dish. Ensure that the batter is distributed as evenly as possible, but refrain from stirring it into the fruit to maintain layers.
Place the baking dish in the preheated oven and bake for 45-50 minutes. The cobbler is done when the top is golden brown and a toothpick inserted into the center comes out clean.
Once baked, remove the cobbler from the oven and let it cool for 5-10 minutes before serving.
Notes
For an inviting presentation, serve warm slices topped with a scoop of creamy vanilla ice cream.