Preheat the oven to 350 and grease an 8x8 baking dish.
In the bowl of an electric mixer (or using a hand held mixer), cream together butter and sugar for a couple of minutes.
Add egg and vanilla and beat until mixed.
Lower the speed to low, sift in the flour, baking soda, and salt. Mix the ingredients on low speed until just combined.
Spread cookie batter in the baking dish, evenly.
Pit and chop peaches into about 1/4 inch chunks. In a mixing bowl, carefully mix peaches, flour and sugar just until peaches are coated. Don't mash the peaches!
Spread peaches over the cookie batter, evenly.
In a mixing bowl, mix flour, cinnamon, brown and white sugar until evenly incorporated. Pour in melted butter and mix it with a fork, forming coarse crumbles.