Butter the inside of each inside muffins and lightly toast under broiler.
Add remaining butter to a non-stick skillet and heat over medium heat. Carefully crack eggs into skillet. Top with a pinch of salt & pepper. Cook until whites are gently set, then flip eggs carefully. Cook to your desired doneness. (We turn off the burner right after flipping for a perfectly runny egg.)
Build sandwiches: Spread mayo on inside of English muffin. Top with cooked bacon and red onion. Carefully slide egg onto the sandwich and immediately top with shredded TomaRashi, allowing the cheese to melt somewhat from the heat of the egg. Top with arugula and the top of the English muffin.
