Put the boneless skinless chicken breast in a medium saucepan and add the water to cover by about 1 inch. Bring just to a bare simmer over medium heat, then reduce heat to maintain gentle bubbling; poach until the thickest part reaches 165°F and the meat is opaque throughout, 12–15 minutes. Transfer to a plate to cool for 10 minutes, then chop into 0.5-inch pieces.
In a large bowl, whisk together the mayonnaise, Dijon mustard, lemon juice, kosher salt, and black pepper until smooth.
Add the chopped celery, red onion, fresh parsley, and the chopped chicken to the bowl. Fold until everything is evenly coated; taste and adjust seasoning if needed.
Toast the sandwich bread slices if desired until lightly golden, 1–2 minutes, as a cue the edges should feel crisp.
Lay out 4 slices of bread, add a leaf or two of lettuce to each, and divide the chicken salad over the lettuce. Cap with the remaining bread slices, press gently, cut if you like, and serve at once.