Slice chicken breasts into strips and season with salt and pepper. In a non-stick skillet, heat oil over medium high heat. Add chicken and cook until golden brown and chicken is no longer pink in the middle, about 4-5 minutes. Remove chicken to paper towels to drain. Combine chicken and sliced boiled potatoes in a large bowl and set aside.
For the dressing, combine garlic, mustard, vinegar and two pinches of salt in a blender and process. Add mayonnaise and blend together to form a thick base. In a slow stream add olive oil through hole in lid. Scrape dressing with spatula into a bowl and season to taste with salt, pepper and lemon juice. If desired, add anchovy to dressing to create a deeper, saltier flavor.
Toss the chicken and potatoes with 3 or 4 tablespoons of the dressing. Place ripped up romaine lettuce leaves in a large bowl and toss with several tablespoons of the dressing. Arrange lettuce on a plate and top with chicken and potatoes. Garnish with chopped parsley and anchovies, if desired.