Boil the chicken breasts in a pot of salted water over medium-high heat for about 15 minutes until fully cooked and tender, then remove and let cool slightly.
While the chicken cools, chop the celery stalks into small, uniform pieces on a cutting board using a sharp knife, ensuring even texture throughout.
Shred the cooled chicken into bite-sized pieces using two forks or your fingers, aiming for a loose, shredded texture.
Combine the shredded chicken and chopped celery in a mixing bowl. Add mayonnaise, lemon juice, Dijon mustard, and season with salt and pepper, mixing thoroughly until evenly coated and creamy.
Toast bread slices until golden brown and slightly crisp, either in a toaster or on a grill, ensuring uniform color and aroma.
Spread the chicken salad mixture evenly onto four slices of toasted bread, then top with remaining slices to form sandwiches. Serve immediately for best texture and flavor.