Step 1: Preheat your oven to 375°F (190°C). Lightly grease a 9x13 inch baking dish.
Step 2: Prepare the peach filling. In a large bowl, combine the sliced peaches, 1 cup granulated sugar, 1 teaspoon vanilla extract, 1/4 cup flour, 1/2 teaspoon cinnamon, and a pinch of nutmeg. Toss gently to coat. Pour the peach mixture into the prepared baking dish and dot with the 2 tablespoons of butter pieces.
Step 3: Prepare the sweet crust dough. In a separate medium bowl, whisk together 1 1/2 cups flour, 1/2 cup granulated sugar, baking powder, and salt. Cut in the cold butter with a pastry blender or your fingertips until the mixture resembles coarse crumbs. Stir in the milk until just combined, being careful not to overmix.
Step 4: Assemble the cobbler. Drop spoonfuls of the dough evenly over the peach filling in the baking dish. If desired, lightly brush the topping with an egg wash (1 egg beaten with 1 tbsp water) and sprinkle with 1 tablespoon of sugar for extra sparkle and crunch.
Step 5: Bake until golden brown. Bake for 35-45 minutes, or until the crust is golden brown and the peach filling is bubbly around the edges. If the crust starts to brown too quickly, you can lightly tent it with aluminum foil.
Step 6: Cool and serve. Remove the cobbler from the oven and let it cool for at least 15-20 minutes before serving. This allows the filling to set slightly. Serve warm, perhaps with a scoop of vanilla ice cream or a dollop of whipped cream.