Preheat the oven to 350°F (175°C). In a large mixing bowl, combine the blackberries, 1/2 cup sugar, and lemon juice. Gently toss until the blackberries are evenly coated, then transfer to the baking dish.
In a separate bowl, whisk together the flour, 1/4 cup sugar, baking powder, and salt. Add the melted butter and stir with a spatula until just combined, creating a thick batter.
Drop spoonfuls of the batter evenly over the fruit filling in the baking dish, covering most of the surface but leaving some gaps for bubbling fruit.
Place the baking dish in the preheated oven and bake for about 40 minutes, or until the topping is golden brown and the blackberry filling is bubbling around the edges. You should hear gentle bubbling when it is ready.
Remove from the oven and let cool slightly before serving. The cobbler should have a deep amber topping with vibrant, juicy blackberries visible through the cracks.