Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.
Step 2: If using fresh peaches, bring a large pot of water to a rolling boil. Carefully drop peaches into the boiling water for 30-60 seconds, then immediately transfer them to an ice bath. The skins should slip off easily. Peel, pit, and slice the peaches into about 1/2-inch thick wedges. You should have approximately 6 cups.
Step 3: In a large bowl, gently combine the sliced peaches with 1/2 cup granulated sugar, brown sugar, cornstarch, ground cinnamon, ground nutmeg, lemon juice, and 1 teaspoon vanilla extract. Toss carefully to coat all the peach slices evenly.
Step 4: Pour the peach mixture into the prepared baking dish. Dot the top of the peach mixture with 2 tablespoons of cold, cubed unsalted butter. Set aside.
Step 5: In a separate medium bowl, whisk together the 1 1/2 cups all-purpose flour, 1/2 cup granulated sugar, baking powder, and salt.
Step 6: Add the 1/2 cup cold, cubed unsalted butter to the dry ingredients. Using a pastry blender, your fingertips, or two knives, cut the butter into the flour mixture until it resembles coarse crumbs with some pea-sized pieces of butter remaining.
Step 7: Gradually add the milk to the flour-butter mixture, stirring just until the dry ingredients are moistened and a soft dough forms. Be careful not to overmix.
Step 8: Drop spoonfuls of the dough evenly over the peach filling in the baking dish. You can leave small gaps between the dough drops, or spread it slightly to cover more surface. Sprinkle the topping with an additional tablespoon of granulated sugar.
Step 9: Place the baking dish on a baking sheet (to catch any potential drips) and bake in the preheated oven for 45-55 minutes, or until the peach filling is bubbly and thick, and the cobbler topping is golden brown and cooked through.
Step 10: Once baked, remove the peach cobbler from the oven and let it cool for at least 15-20 minutes before serving. This cooling time allows the juices to set and thicken properly.
Step 11: Serve warm, ideally with a scoop of vanilla bean ice cream or a dollop of freshly whipped cream. Enjoy!