Preheat the oven to 375°F. Grease a 9×13-inch baking dish or a large cast iron skillet.
In a large bowl, combine the sliced peaches, sugar, lemon juice, vanilla, cinnamon, nutmeg (if using), and arrowroot powder. Toss gently to coat.
Pour the peach mixture evenly into the prepared baking dish.
In a separate bowl, whisk together the flour, sugar, baking powder, baking soda, and salt.
Cut the cold butter into the dry ingredients using a pastry cutter or your fingers until the mixture resembles coarse crumbs.
Stir in the sourdough discard and milk just until combined. The dough should be thick and spoonable.
Drop spoonfuls of the topping evenly over the peaches, leaving small gaps for the filling to bubble through.
Bake uncovered for 40–45 minutes, or until the topping is golden and the peach filling is bubbling around the edges.
If the topping browns too quickly, loosely cover with foil during the last 10 minutes of baking.
Remove from the oven and let cool for 10–15 minutes before serving.