Combine peaches, 1 cup white sugar, and water in a saucepan; bring to a boil. Reduce heat and simmer, stirring carefully, until peaches are tender and syrup is thickened, about 10 minutes.
Put butter in a 3-quart baking dish and place in oven until butter melts, 2 to 3 minutes.
Mix flour, milk, brown sugar, and remaining 1/2 cup white sugar together in a bowl until batter is smooth; pour over melted butter without stirring. Spoon peach mixture over batter, allowing the syrup to drip over peaches into the batter. Sprinkle cinnamon over the top.
Bake in the preheated oven until cobbler is cooked through and bubbling, about 45 minutes.