Preheat your oven to 375°F (190°C).
Bring a large pot of water to boil. Score an "X" on the bottom of each peach, then blanch them for 30-45 seconds. Transfer immediately to an ice bath. The skins should now slip off easily with your fingers or a paring knife.
Slice peaches into 1/2-inch wedges, discarding the pits.
In a large bowl, combine the sliced peaches with granulated sugar, brown sugar, flour, cinnamon, and nutmeg. Add the lemon juice and vanilla extract, then gently toss until the peaches are evenly coated.
Transfer the peach mixture to a 9x13 inch baking dish, spreading it evenly. Dot the top with the small pieces of butter.
In a medium bowl, combine flour, granulated sugar, brown sugar, baking powder, and salt. Using a pastry cutter or two forks, cut in the cold cubed butter until the mixture resembles coarse meal with pea-sized pieces remaining.
Pour the boiling water over the flour mixture and stir just until combined.
Drop spoonfuls of the dough evenly over the peach filling, leaving some spaces for the filling to bubble through.
Sprinkle the cinnamon-sugar mixture over the top.
Bake for 45-50 minutes, or until the topping is golden brown and the filling is bubbling around the edges.
Allow the cobbler to cool for at least 15 minutes before serving to let the filling thicken.