Preheat the oven to 375 F. Lightly grease a 9x13-inch baking dish and set it aside.
Add the peaches and blackberries to a large bowl. Add in 1 cup of sugar, ¼ cup of flour, cinnamon, lemon juice, and lemon zest. Gently mix until the fruit is coated.
In the bowl of a food processor add the remaining ¼ cup of sugar, 1 ½ cup of flour, salt, and baking powder. Pulse for a few seconds until everything is mixed well. Add in the butter and continue pulsing until the mixture becomes coarse crumbs. Pour into a separate medium bowl.
Add the milk and vanilla extract to the medium bowl and stir until just combined.
Pour the fruit mixture into the bottom of the prepared baking dish. Place dollops of the batter on top of the fruit mixture.
Bake the cobbler in the preheated oven for 45 to 55 minutes, until the top is golden brown and the fruit is bubbly.