Step 1: Preheat your oven to 375°F (190°C). Grease a 9x13 inch baking dish.
Step 2: In a large bowl, combine the sliced peaches, 1/2 cup granulated sugar (for filling), cornstarch, cinnamon, nutmeg, and lemon juice. Toss gently to coat. Let the mixture sit for 10-15 minutes to allow the juices to release.
Step 3: While the peaches are sitting, prepare the topping. In a separate medium bowl, whisk together the all-purpose flour, 1/2 cup granulated sugar (for topping), baking powder, and salt.
Step 4: Cut the cold butter into the dry ingredients using a pastry blender or your fingertips until the mixture resembles coarse crumbs. Do not overmix.
Step 5: Add the milk and vanilla extract (if using) to the flour-butter mixture. Stir just until the dry ingredients are moistened and a soft dough forms. Be careful not to overmix.
Step 6: Pour the peach filling into the prepared baking dish, spreading it evenly.
Step 7: Drop spoonfuls of the cobbler topping over the peach filling. Don't worry about covering the entire surface; the rustic look is part of its charm. If desired, sprinkle with coarse sugar.
Step 8: Bake for 35-45 minutes, or until the topping is golden brown and the peach filling is bubbly. If the topping starts to brown too quickly, you can loosely tent the dish with aluminum foil.
Step 9: Remove from oven and let cool for at least 15-20 minutes before serving. This allows the filling to set slightly.