The breakfast sandwich can, and often should, be as simple as possible. My longtime go-to has been this recipe which keeps it as simple as possible with a slice of prosciutto, an egg, some butter, and an English muffin. Why introduce cheese or tomato or anything else when it tastes that good?
That said, I usually listen to Nancy Silverton, and especially to her sandwich cookbook, where I got this recipe. First, she serves it open-faced on crusty bread, which is rubbed with a garlic clove, Spanish-style, adding the intoxicating hint of garlic but nothing more. The real story here might be the scallion oil, which takes about ten seconds to make and is nothing more than scallions and parsley pureed with olive oil.