In a medium bowl, whisk the mayonnaise, butter, lemon juice, and a pinch each of salt and pepper until smooth.
Coat a large nonstick skillet with a thin sheen of butter over medium heat. Add the eggs and cook until the bottoms are just set and not browned, about 1 minute. Carefully fold half of each egg white over the yolk to enclose it, forming a half moon shape. Cook until the whites are just set, but the yolks are still runny, about 2 minutes.
Place 1 slice of ham on each English muffin half. Top with an egg and the sauce.