Preheat your oven to 350°F (175°C).
In a large mixing bowl, combine the blackberries and raspberries. Add the lemon zest and 2 tablespoons of granulated sugar. Gently toss the berries to coat them with sugar and zest, allowing them to release some of their juices. Set aside.
In another bowl, whisk together the self-rising flour, 1 cup of granulated sugar, baking powder, and salt.
In a separate bowl, mix the melted butter and milk until well combined. Gradually add this mixture to the dry ingredients, stirring until just combined.
Pour the berry mixture into the bottom of your greased baking dish. Spoon the batter over the berries, spreading it evenly.
Bake in the preheated oven for about 40-45 minutes, or until the top is golden and a toothpick inserted into the center comes out clean.
Once baked, remove the cobbler from the oven and let it cool for a few minutes before serving.