Preheat your oven to 375°F (190°C). Lightly grease a 9-inch square or similarly sized baking dish.
Peel and slice the peaches. Place them in a bowl and gently toss with sugar, lemon juice, vanilla extract, cinnamon, and flour or cornstarch. Allow the peaches to sit for about 5–10 minutes so they can release some of their natural juices.
In a medium bowl, whisk together the flour, baking powder, and salt.
In another bowl, combine the melted butter, sugar, milk, and vanilla extract. Gradually add the dry ingredients and stir just until combined. Avoid overmixing, as this can make the topping dense.
Spoon the peach mixture evenly over the batter. Do not worry if the peaches do not form a perfectly even layer—the batter will rise around the fruit as it bakes.
Sprinkle the remaining 1/2 cup of granulated sugar evenly over the peach and batter mixture.
Although it may look unusual at this stage, this technique is what creates the signature sweet crust as the cobbler bakes.
Place the dish in the preheated oven and bake for approximately 45–50 minutes, or until the top is golden brown and the edges are bubbling.
The crust should look golden and slightly crisp while the peach filling should be bubbling underneath.
Allow the cobbler to rest for approximately 10–15 minutes before serving. This gives the peach juices a chance to thicken slightly.
Serve warm with vanilla ice cream, whipped cream, or a spoonful of lightly sweetened cream.