Preheat the oven to 350°F.
Combine the peaches, 1 cup sugar, and water in a saucepan. Mix well, bring to a boil, then reduce the heat and simmer for 10 minutes. Remove from the heat.
Place the butter in a 3-quart baking dish and put the dish in the oven until the butter melts.
In a mixing bowl, slowly combine the remaining 1 cup sugar, self-rising flour, and milk. Stir gently until smooth and avoid overmixing.
Pour the batter over the melted butter in the baking dish. Do not stir.
Spoon the peaches over the batter, then gently pour the syrup over the top. Sprinkle with ground cinnamon if using.
Bake for 30 to 45 minutes, until the top is golden and the center looks set.
Let the cobbler rest for 10 to 15 minutes before serving. Serve warm with whipped cream or vanilla ice cream.