Position a rack in the middle of the oven and heat to 375°F (190°C). Lightly grease a 9×13-inch baking dish with a bit of butter or cooking spray.
In a large bowl, toss the sliced peaches with 1/3 cup granulated sugar and the cinnamon. If you want a thicker filling, stir in the cornstarch. Let the fruit sit while you make the topping — 10 to 15 minutes will help the peaches release their juices.
Transfer the macerated peaches and any accumulated juices to the prepared baking dish and spread them into an even layer.
In a medium bowl, whisk together the flour, baking powder, salt and 2/3 cup granulated sugar. Use a fork to combine the dry ingredients until evenly mixed.
Pour the melted butter and vanilla over the dry mixture. Stir with a wooden spoon or spatula until a shaggy, slightly sticky dough forms — it should resemble a thick drop-biscuit batter rather than a smooth cake batter.
Drop the topping by tablespoonfuls over the peaches, spacing the dollops 1 to 2 inches apart. Use the back of a spoon or a small offset spatula to gently spread and join some pieces — you want an uneven, rustic surface so juices can bubble up around the dough.
Sprinkle the turbinado sugar evenly over the top if using. Bake the cobbler, uncovered, for 30 minutes at 375°F. The filling should be bubbly and the topping mostly set.
Increase the oven temperature to 425°F and continue baking 8 to 10 minutes more, or place under the broiler for 2–4 minutes just until the topping develops golden-brown edges and a bit of crunch. Watch closely to prevent burning.
Remove from the oven and let the cobbler rest for at least 15–20 minutes so the juices thicken slightly. Serve warm with scoops of vanilla ice cream or a dollop of whipped cream.