Mix mayonnaise, parmesan, lemon juice, and pesto. Spread pesto on one side of sandwich bread. Spread butter on other side.
Poach 24 fresh eggs for 5 minutes in 2 quarts simmering water with 1/3 cup vinegar. Place cooked eggs on paper towels to drain and cool.
Place 2 eggs on slice of bread and make sandwiches. Can be made a few hours ahead but must be kept cool.