Step 1

Adjust oven rack 5 inches from broiler element and heat broiler. Spread insides of muffins evenly with butter and arrange split side up on rimmed baking sheet. Combine mayonnaise and mustard in bowl; set aside. Crack 2 eggs into small bowl and season with salt and pepper. Repeat with remaining 2 eggs and second small bowl.

Step 2

Broil muffins until golden brown, 2 to 4 minutes, rotating sheet halfway through broiling. Flip muffins and broil until just crisp on second side, 1 to 2 minutes; set aside while cooking eggs.

Step 3

Heat oil in 12-inch nonstick skillet over medium-high heat until shimmering. Working quickly, pour 1 bowl of eggs in 1 side of pan and second bowl of eggs in other side. Cover and cook for 1 minute.

Step 4

Working quickly, top each egg with 1 slice of ham and ¼ cup Gruyère. Cover pan, remove from heat, and let stand until Gruyère is melted and egg whites are cooked through, about 2 minutes.

Step 5

Spread mayonnaise mixture on muffin bottoms and place 1 ham-and-cheese-topped egg on each. Divide arugula evenly among sandwiches, then top with tomato slices and muffin tops. Serve.