Preheat your oven to 375°.
Combine the sliced peaches, brown sugar, granulated white sugar, cinnamon, nutmeg, cornstarch, and lemon juice in a medium bowl. Stir it and then pour it into a 9x9 in. baking dish.
Cover the baking dish with foil and bake for 20 minutes.
While it is baking, cream the softened butter and granulated white sugar for 4 minutes in a medium sized mixing bowl.
Add in the egg and vanilla extract and mix for 4 minutes.
Gradually add in the flour, baking powder, and salt until just combined.
Once the peach filling is done, remove the foil and drop the sugar cookie topping by large spoonfuls evenly across the top of the peaches.
Put it back in the oven and bake for 40-50 minutes, or until it is golden brown and bubbling. You just don't want any gooey cookie dough in the middle...although, maybe you do!:)