Prepare a charcoal or gas grill for indirect heat: If using a gas grill, turn all burners to high and close the lid. When the temperature inside the grill reaches medium high heat or about 400°, lift lid and turn off half of the burners. The area with no active burners is the indirect heat area. If using a charcoal grill, light a chimney full of briquettes and let burn until covered with ash, about 15 to 25 minutes. Mound them to one side. The area over the section cleared of coals is the indirect heat area. Thoroughly clean grates.
In a large bowl, combine the peaches, zest, sugar, corn starch and salt until the peaches are evenly coated. Add berries and toss to gently to combine. Set aside while you prepare the topping. Pour melted butter into a 12-inch cast iron skillet.
For the Topping
In a medium bowl, whisk together flour, sugar, baking powder, and salt. Add cold butter and toss to break up the pieces, then smash each one flat between your fingertips. Continue smashing and rubbing until butter is broken up into small pieces like cereal flakes. Stir in buttermilk and vanilla with a flexible spatula, folding to create a dough.
Pour the peach mixture into the buttered skillet and drop the thick dough in 2-tablespoon portions over the fruit spaced evenly apart. Place on the grill over indirect heat and cook until the biscuits are golden brown and the fruit is bubbling, checking every 15 minutes. The cobbler will cook for about 1 hour. Cool at least another hour before serving, to allow the filling to thicken (while piping hot, it will be runny and thin).