1. Combine first 6 ingredients in a medium bowl, stirring well.
2. Cut 2 eggs in half lengthwise; reserve 2 yolks for another use. Coarsely chop egg whites and whole eggs. Add eggs to mayonnaise mixture; stir gently to combine.
3. Arrange 4 toast slices on a cutting board or work surface. Top each toast slice with 1 lettuce leaf, 1/2 cup egg mixture, and 1 toast slice. Serve immediately.