Preheat a cast iron or heavy bottomed stainless steel skillet to low-medium heat for 3-5 minutes, or until heat radiates from the bottom of the pan.
Slice your sourdough bread into slices that are ½-3/4" thick.
Spread softened butter on one side of each of your slices of sourdough bread and set aside, unbuttered side down.
Once the skillet has been preheated, add one slice of sourdough bread to the pan, buttered side down.
Top the bread with the shredded cheese, aiming to keep it on the bread, although it's okay if a few shreds fall down onto the pan.
Top the shredded cheese layer with the other slice of sourdough bread, with the buttered side facing up (towards you).
Cook for 2-3 minutes, or until the bottom is toasty and crisp. If you have a lid for your skillet, place it on top for 1-2 minutes to help melt the cheese.
Gently flip the sandwich to the other side, using your hand to hold the top slice in place as you flip.
Cook for an additional 2-3 minutes, or until the cheese is mostly melted and the bottom slice is toasty brown.
Remove the sourdough grilled cheese from the skillet and place on a cutting board.
Use a serrated knife to slice the sandwich in half and serve immediately.