Heat the oven to 375°F (190°C). Lightly butter a 2-quart baking dish. Toss the peaches with 1/4 cup sugar, lemon juice, cinnamon, and cornstarch. Spread them in the dish.
Whisk the flour, baking powder, salt, and remaining 1/4 cup sugar in a bowl. Cut in the cold butter until the mixture has pea-size pieces.
Stir the milk and vanilla into the flour mixture, then fold just until a soft dough forms. Drop spoonfuls of dough over the peaches, leaving small gaps for steam to escape.
Bake for 35 to 40 minutes, until the topping is golden and the peach filling bubbles around the edges.
Cool the cobbler for at least 15 minutes before serving so the peach juices can thicken.
