Put mushrooms in large skillet with a splash of water. Cook over medium heat until mushrooms shrink and give up most of their liquid, about 5 minutes. Add spinach, scallions, oregano, salt, pepper, and nutmeg, and cook just until spinach wilts and liquid is gone, 3 to 4 minutes. Remove to a plate and wipe out skillet with a paper towel.
For each sandwich, put a thin layer of cheese on a slice of bread. Top with 1/4 of the mushroom mixture, another thin layer of cheese and the other slice of bread. Press down gently to compact sandwich.
Working in batches, heat oil in skillet over medium heat. When hot, add sandwiches and cook until lightly browned and cheese melts, 2 to 3 minutes per side. Remove and cool slightly, then cut on a diagonal.
Tip: Cook the sandwich on a panini press if you like.