Preheat your oven to 350°F.
In a large bowl, combine the sliced peaches, 1/2 cup of granulated sugar, lemon juice, vanilla extract, and cinnamon. Toss until the peaches are well coated.
Pour the peach mixture into a greased 9x13-inch baking dish.
In another bowl, whisk together the flour, baking powder, salt, and remaining 1/2 cup of granulated sugar.
In a separate bowl, mix the melted butter, buttermilk, and egg until well combined.
Gradually add the wet ingredients to the dry ingredients, stirring until just combined.
Drop spoonfuls of the batter over the peaches, covering them as evenly as possible.
In a small bowl, mix the brown sugar and nutmeg, then sprinkle it over the batter.
Bake for 40-45 minutes, or until the top is golden brown and a toothpick inserted in the center comes out clean.
Allow the cobbler to cool for about 15 minutes before serving.