Add the oats, milk, grated apple and salt to a large bowl (microwave-safe if you want to serve warm) and stir to evenly combine. Cover and refrigerate overnight.
To make caramelized apple topping, melt butter in a cast iron skillet over medium heat. Add apple and brown sugar and cook until apple is starting to caramelize on the bottom, about 2 minutes. Flip apples, add the water and cover until soft, about 5 minutes. Add more water if the pan gets dry before the apples are soft. Refrigerate until ready to use.
Serve overnight oats cold from the fridge, or microwave 1 minute before topping with apple.