In a deep sauce pan on high heat, brown the stick of butter. Butter should bubble and be a golden brown on the bottom of the pan. Butter burns easily so watch carefully and stir occasionally to ensure even cooking.
Once the butter is completely browned, lower heat and stir in the sugar until it is completely dissolved.
Stir in the peaches with their syrup, salt, cinnamon (if using) and simmer for about 10 minutes.
Dissolve the cornstarch in the water and add to the fruit mixture. Cook for one more minute to thicken.
Remove from the heat and stir in the vanilla.
Make the Batter:
While the peaches are cooking, preheat the oven to 350℉
In a medium mixing bowl or stand mixer, whisk the egg, milk, and vanilla together until incorporated.
Slowly add the dry ingredients to the egg mixture a little bit at a time and beat on medium speed for 2 minutes once all ingredients are added. The batter will be very thin.
Prepare the dish and assemble the cobbler:
Brown the additional stick of butter as directed above and pour in to the bottom of a 2 qt casserole dish.
Pour the batter on top of the butter. Do not stir.
Spoon the peach mixture on top of the batter. Do not stir.
Bake for 30-45 minutes or until the top is golden brown and a toothpick inserted into the cake part comes out clean.