Bring a pot of water to a steady boil over medium-high heat. Using a slotted spoon, add the eggs and cook for 6 1/2 minutes. Remove and place in a bowl of ice water for 2 minutes. Remove from water and carefully peel.

Slice the bagels in half and toast, then butter each side and arrange on a plate. Slice each egg in half and place on a buttered bagel, top with anchovies and season to taste with salt and pepper. Top with green onions if desired.