Preheat oven to 425 degrees F (220 degrees C).
In a large bowl, combine peaches, 1/4 cup white sugar, 1/4 cup brown sugar, 1/4 teaspoon cinnamon, nutmeg, lemon juice, and cornstarch. Toss to coat evenly, and pour into a 2 quart baking dish. Bake in preheated oven for 10 minutes.
Meanwhile, in a large bowl, combine flour, 1/4 cup white sugar, 1/4 cup brown sugar, baking powder, and salt. 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 topping over them. Sprinkle entire cobbler with the sugar and cinnamon mixture. Bake until topping is golden, about 30 minutes.
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