Combine first 6 ingredients in a medium bowl, stirring well.
Coarsely chop eggs. Add eggs to mayonnaise mixture; stir gently to combine.
Arrange 4 bread slices on a cutting board or work surface. Top each bread slice with 1/2 cup egg mixture, 2 bacon pieces, 2 tomato slices, 1 lettuce leaf, and 1 bread slice. Serve sandwich immediately.