Wash and slice the peaches, then place them into a mixing bowl.
Add the brown sugar, cinnamon, nutmeg, and orange juice to the peaches.
Stir everything together until the peaches are fully coated, then set the bowl aside.
Add a stick of unsalted butter to a 9-inch cast-iron skillet.
Place the skillet into the oven and preheat the oven to 375 degrees Fahrenheit, allowing the butter to fully melt as the skillet heats up.
While the oven preheats, whisk together the flour, white sugar, kosher salt, and baking powder in a mixing bowl.
Pour in the milk and vanilla extract and whisk until everything is fully combined and smooth.
Carefully remove the hot skillet from the oven and pour the batter directly into the melted butter, but don't stir it.
Place the skillet back into the oven and bake the batter for about 5 minutes.
Remove the skillet from the oven and spoon the peaches and all of their juices evenly across the top. Again, don't stir.
Bake the peach cobbler for about 30 to 40 minutes, or until the center is set and the top is golden brown.
Once baking is complete, remove the cobbler from the oven and let it rest for about 5 to 10 minutes before serving.