Preheat oven to 350°F.
Place stick of butter (see crust ingredients) in an 11x7 inch or 9x9 inch baking dish. Heat in the oven for 5-10 minutes, just until butter melts. Very carefully remove dish from oven and set aside.
Create constarch slurry by mixing cornstarch and cool water in a small bowl until they are smooth.
To a large mixing bowl, add peaches with syrup, granulated sugar, brown sugar, salt, ground cinnamon, and ground nutmeg. Mix with spoon until sugars are dissolved and spices are incorporated.
Add lemon juice, vanilla extract, and cornstarch slurry. Mix until well combined. Set peach filling mixture aside.
In a separate mixing bowl, make batter for the crust by combining all-purpose flour, baking powder, brown sugar, granulated sugar, and salt. Whisk together until well combined and lumps of brown sugar are broken down.
Slowly add cold milk to dry ingredients, about 1/3 at a time, and mix well after each addition.
Pour the batter into your baking dish, on top of the melted butter. Gently spread batter to the edges of the baking dish, but do not stir the butter and batter together.
Evenly place peaches and liquid on top of batter. Lightly sprinkle the top of cobbler with additional ground cinnamon, if desired.
Place the baking dish on top of a foil-lined cookie sheet to catch any juice that bubbles over. Bake in preheated oven for 45-55 minutes, or until cobbler is golden brown and bubbly & crust is cooked through. Cool in dish for at least 20 minutes before serving.