In a bowl, whisk the eggs lightly. Add the 1 tablespoon mayonnaise, the crab, breadcrumbs, 2 teaspoons salt, ½ teaspoon pepper and green onions. Stir with a fork until well mixed. Divide the mixture into 8 equal portions and gently shape into patties.
In a small bowl, whisk together the 1/3 cup mayonnaise, the mustard, lemon juice and ½ teaspoon pepper until smooth.
In a large frying pan, melt the butter over medium-low heat. Working in batches if needed, add the crab cakes and cook without moving them until golden brown on the first side, about 4 minutes. Turn and cook until golden brown on the second side, 3 to 4 minutes. Arrange the lettuce on plates and place the crab cakes on top. Drizzle with your favorite creamy mustard dressing and serve.