Step 1. Over medium heat, melt 2 tablespoons butter in a large frying pan. Add sliced onion and adjust heat to medium-low. Cook until onion is soft and brown, around 30 minutes, stirring every once in a while.
Step 3. Using same skillet, adjust heat to low and add remaining 2 tablespoons of butter to pan. Place two slices of sourdough in pan, and distribute cheese evenly over both slices of sourdough. Add onion mixture and place remaining 2 slices of sourdough on top.