Place a rimmed baking sheet lined with foil in the oven and preheat to 350°F. Lightly grease a 9 x 11-inch rectangular baking dish or 11-inch oval baking dish.
Make peach filling: Whisk together sugar, cornstarch, and salt in a large bowl. Add sliced peaches, lemon zest, and lemon juice. Toss gently to combine. Spoon peach filling into prepared baking dish.
Make cobbler topping: Whisk together flour, sugar, baking powder, and salt in a medium bowl. Add melted butter, sour cream, and vanilla and fold together until fully incorporated and a cookie-like dough forms. Drop topping by spoonfuls over the peaches and sprinkle with turbinado sugar.
Bake cobbler: Place baking dish on preheated baking sheet and bake for 60 minutes, or until golden and filling is bubbling. For an extra golden top, move baking dish to the top rack of the oven for the last 5 minutes of baking. Cool cobbler on a wire rack for 1 hour before serving.
Store leftover cobbler tightly covered (or in an airtight container) in the refrigerator up to 2 days. Reheat in a warm oven (325°F) for 15 minutes, until heated through.