Preheat the oven to 350°F. Line a baking sheet with foil, which will catch the drips from the cobbler. Place the sliced peaches into a 9"x13" baking dish, then add the lemon zest and juice on top.
In the bowl of a stand mixer, beat together the butter, 1 cup of the white sugar, and the brown sugar, until fluffy. Add the flour and baking powder until the mixture is evenly sandy. Turn the mixer to low and slowly add in the milk and the vanilla, mixing for an extra 2 minutes until the batter is light.
Drop dollops of the batter evenly over the top of the peaches in the dish and, using an offset spatula, smooth the batter out so that the peaches are barely visible and the batter is no more than ½ inch thick in any place.
Sprinkle the remaining white sugar over the batter evenly, then drizzle the hot water on top. Bake for 60 to 70 minutes, until the top is crackled and golden.
Allow to sit for 30 minutes so that the filling can thicken, then serve warm. YUM!