Toss peaches with brown sugar, white sugar, cinnamon, nutmeg, lemon juice, and cornstarch in a 2 qt nonreactive baking dish. Bake in middle of the oven for 10 minutes.
While peaches bake, make topping by stirring together flour, sugars, 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. Then sprinkle entire cobbler with the cinnamon & sugar mixture. Bake in middle of oven until topping is golden, about 30 minutes.