In a large saucepan, cover the chicken breasts with water. Bring to a very slow simmer and cook over low heat until white throughout, about 18 minutes. Transfer the chicken to a plate and let stand until cool enough to handle, about 15 minutes. Cut the chicken into 1/2-inch dice.
In a large bowl, mix the mayonnaise with the mustard and season with salt and pepper. Fold in the onion, celery, parsley and chicken until evenly coated. Cover and refrigerate until chilled, about 30 minutes.
Spoon the chicken onto 6 slices of the toast. Top with the lettuce and tomato, close the sandwiches and serve.