Heat the oven: Preheat to 350°F (175°C). Place butter in the baking dish and set it in the oven to melt. Remove as soon as melted; keep the dish buttered and warm.
Prepare peaches: In a saucepan, combine peaches, sugar, lemon juice, vanilla, cinnamon, nutmeg, cornstarch, and a pinch of salt. Cook over medium heat 3–5 minutes until the fruit releases juices and the mixture looks glossy and slightly thickened. If using canned peaches, heat just until warm.
Make the batter: In a bowl, whisk flour, sugar, baking powder, and salt. Add milk and vanilla; whisk until smooth. Do not overmix.
Assemble: Pour the batter evenly over the melted butter in the warm dish—do not stir. Spoon the warm peaches and their juices over the batter, distributing evenly. The batter will rise around the fruit as it bakes.
Bake: Bake 40–50 minutes until the top is deep golden, the edges are crisp, and the filling is bubbling. A tester inserted into the batter portion should come out with moist crumbs.
Rest & serve: Cool 15–20 minutes to set. Serve warm with vanilla ice cream or lightly sweetened whipped cream for the most nostalgic Old Fashioned Peach Cobbler experience.