Preheat the oven to 375°F. Lightly butter a 9x13-inch baking dish and set it aside.
In a large bowl toss the sliced peaches with 1/3 cup granulated sugar, the cornstarch (if using), and the ground cinnamon. Let the mixture sit and macerate for 10–15 minutes so the peaches release some juice.
While the peaches macerate, whisk together the flour, baking powder and salt in a medium bowl. Stir in the 2/3 cup sugar until evenly distributed.
Add the melted butter and vanilla to the dry mixture and stir until a thick, slightly sticky dough forms. The texture should be more like a stiff biscuit dough than a pourable batter.
Pour the peach mixture and all accumulated juices into the prepared pan, spreading into an even layer.
Drop tablespoons of the topping over the peaches, spacing them out so some fruit peeks through. If needed, gently press a few dollops to help cover the fruit. For neater dollops, chill the mixed topping 5–8 minutes before scooping.
Bake the cobbler in the preheated oven for 35–40 minutes, or until the fruit is bubbly at the edges and the topping is set and golden brown.
If you want a crisper, caramelized top, move the pan under the broiler for 1–3 minutes—watch carefully so it doesn't burn.
Remove from the oven and let rest 20–30 minutes so the filling thickens. Serve warm with a scoop of vanilla ice cream or a dollop of whipped cream.