Combine peaches, 2 tablespoons brown sugar, 1/4 teaspoon cinnamon, nutmeg, lemon juice, and cornstarch in a large bowl. Toss to coat evenly, and pour into a 2-quart baking dish.
Bake in the preheated oven for 10 minutes.
Meanwhile, combine flour, white sugar, 1/4 cup brown sugar, baking powder, and salt in a large bowl. Blend in butter with your fingertips or a pastry blender until mixture resembles coarse meal. Stir in water until just combined.
Remove peaches from oven, and drop spoonfuls of flour mixture over them.
Mix together 2 tablespoons brown sugar and 1 teaspoon ground cinnamon. Sprinkle entire cobbler with the sugar and cinnamon mixture. Bake until topping is golden, about 30 minutes.
