Slice the mushrooms and onion and shred the cheese.
Melt 4 tablespoon butter in a large skillet over high heat until foaming then add onions and mushrooms. Season with garlic powder, salt, and pepper, and leave undisturbed for 1 minute.
Cover and cook 3-5 min, stirring a couple of times, until the onions have softened.
Uncover, reduce heat to medium, and continue stirring occasionally for about 30 more minutes, until the mushrooms and onions are golden brown. If the bottom of the skillet starts to dry, add a couple tablespoons of water and scrape up any stuck bits.
Once caramelized, reduce heat to low and pour in the soy sauce to deglaze the skillet before adding the arugula, miso paste, and shredded cheese. Stir for a couple of minutes, until the cheese has slightly melted and the arugula has started wilting.
Scoop the mushroom mixture onto three slices of bread. Butter one side of three more slices of bread. Place the buttered side up on top of each mushroom slice and push down to create a sandwich.
Melt 2 tablespoon of butter in the skillet over medium heat. Place the sandwiches buttered side down and cook 2-4 min, until golden brown. Keeping in the skillet, butter the side facing up, add the remaining 1 tablespoon butter, and flip the sandwiches. Gently press down and cook until the other side is golden brown. Serve immediately.