Preheat your oven to 375°F and butter a 9x13-inch baking dish. Toss the sliced peaches with sugar, cornstarch, vanilla, cinnamon, and salt in a large bowl until evenly coated. Let this sit for 10 minutes to allow peaches to release their natural juices.

In a separate bowl, whisk together flour, sugar, baking powder, and salt. Cut in the cold butter using a pastry cutter or two forks until the mixture resembles coarse crumbs with some pea-sized butter pieces remaining. In a small bowl, whisk the milk and egg together, then pour into the flour mixture. Stir just until combined; the dough should look shaggy and slightly sticky.

Pour the seasoned peaches and their accumulated juices into your prepared baking dish, spreading them evenly. Drop spoonfuls of the cobbler dough over the peaches, leaving some gaps for a rustic appearance.

Bake for 40-45 minutes until the topping is golden brown and sounds hollow when gently tapped, while the peach filling bubbles vigorously around the edges. If the top browns too quickly, tent loosely with foil during the last 10 minutes.

Let the cobbler rest for 15-20 minutes before serving to allow the filling to thicken slightly and prevent burns from hot fruit.