Step 1: Preheat your oven to 375°F (190°C). Place the stick of unsalted butter into a 9x13 inch baking dish and put it in the preheating oven to melt.
Step 2: In a large bowl, combine the sliced peaches, 1/2 cup granulated sugar, cornstarch, vanilla extract, ground cinnamon, and a pinch of ground nutmeg. Gently toss to coat.
Step 3: In a separate medium bowl, whisk together the all-purpose flour, 1 cup granulated sugar, baking powder, and salt.
Step 4: Pour the milk into the dry ingredients and stir until just combined. Do not overmix.
Step 5: Carefully remove the hot baking dish with the melted butter from the oven. Pour the prepared peach mixture evenly over the melted butter. Do not stir.
Step 6: Spoon dollops of the cobbler topping mixture over the peaches. You don't need to spread it perfectly.
Step 7: Bake for 35-45 minutes, or until the topping is golden brown and the peach filling is bubbly and tender. If the topping browns too quickly, loosely tent with foil.
Step 8: Remove from oven and let cool for 10-15 minutes before serving to allow juices to thicken.