Preheat your oven to 375°F (190°C) and grease a 9x13 inch baking dish.
In a large mixing bowl, combine the sliced peaches, 1/2 cup of sugar, lemon juice, cinnamon, and vanilla extract. Toss until the peaches are well coated and let them sit for about 10 minutes.
In a separate bowl, whisk together the flour, remaining 1/2 cup of sugar, baking powder, salt, and nutmeg if using.
Slowly add the milk to the dry ingredients while mixing until just combined. Do not overmix; a few lumps are fine.
Pour the melted butter into the baking dish, allowing it to coat the bottom. Then immediately pour the batter over the melted butter—do not stir.
Gently spoon the peach mixture on top of the batter. The peaches will sink as it bakes, creating a beautiful layer.
Bake in the preheated oven for 35-40 minutes, or until the cobbler is golden brown and a toothpick inserted in the center comes out clean.
Allow the cobbler to cool slightly before serving.
Notes
Serve warm with a scoop of vanilla ice cream on top and a sprinkle of cinnamon for added flair. Garnish with a sprig of mint for freshness.