Preheat oven to 400 degrees F. and butter an 8-inch square baking pan.
Pit nectarines and slice thin. In a bowl stir together sugar and cornstarch and stir in nectarines, lemon juice, and ginger until combined well. Transfer nectarine mixture to pan.
Into a bowl sift together flour, baking powder, and salt. Cut butter into bits and blend into flour mixture until mixture resembles coarse meal. Stir in milk until just combined and drop topping in small mounds over nectarines. In a small cup stir together sugar and cinnamon and sprinkle over dough.
Bake cobbler in middle of oven 20 to 25 minutes, or until top is golden.