Bring a large pot of water to boil. Cut a small "X" at the bottom of each peach, then place them in the boiling water for 30-60 seconds. Transfer immediately to an ice bath. The skins should now easily slip off. Remove pits and slice peaches into 1/2-inch wedges.
In a large bowl, combine sliced peaches, both sugars, flour, cinnamon, nutmeg, lemon juice, and vanilla. Gently toss until peaches are evenly coated. Let stand for 10 minutes to allow the natural juices to release.
Preheat your oven to 375°F (190°C). Pour the peach mixture into a 9x13 inch baking dish. Dot the top with the small pieces of butter.
In a medium bowl, whisk together flour, both sugars, baking powder, and salt. Using a pastry cutter or your fingertips, work the cold cubed butter into the dry ingredients until the mixture resembles coarse meal with some pea-sized pieces remaining.
Add the boiling water to your flour mixture and stir just until combined. The dough will be sticky and somewhat wet. Using a spoon or your fingers, drop the dough in cobbles over the peach filling, leaving some gaps for the filling to bubble through.
Bake in the preheated oven for 45-50 minutes, until the topping is golden brown and the filling is bubbling around the edges. If the top browns too quickly, loosely cover with aluminum foil.
Allow the cobbler to cool for at least 15 minutes before serving. This cooling period allows the filling to set slightly.