Peel garlic by smashing with the flat side of a knife on a work surface. Finely chop garlic, then add ½ teaspoon salt and anchovy paste. Mash chopped garlic with the flat side of a large, sharp knife until a paste forms.

Shake the oil, lemon zest and juice, anchovies, garlic paste, ½ tsp. salt, and ¼ teaspoon pepper in a jar with a lid.

Put the potatoes in a 3-quart saucepan. Add water to cover and 2 teaspoons kosher salt. Bring to a boil over high heat and cook until tender, about 8-10 minutes. (Exact time will depend on the size of your potatoes.) Drain potatoes well and then transfer to a medium size bowl. Add 2 tablespoons of the dressing, ¼ teaspoon salt and ¼ teaspoon pepper. Stir to coat. (Dressing and potatoes can be made early in the day. Toss the potatoes with the dressing and seasonings, then after they cool down, store in a covered container in the refrigerator. Bring to room temperature or warm a bit in the microwave before serving.)

Toss the chicken in a medium bowl with 2 tablespoons of the dressing ½ teaspoon salt, and ⅛ teaspoon pepper. Allow to marinate for 30 minutes. 

Prepare a medium-high gas or charcoal grill fire.

Arrange the chicken on the grill. Cover the grill and cook, turning the chicken after about 5 minutes, until the chicken is just cooked through, 10 to 12 minutes total. For perfectly juicy chicken, use an instant-read thermometer. Internal temp should be right at 160˚F. Let the chicken rest 5 minutes before slicing.

Meanwhile, in a medium bowl, combine the cucumber, tomatoes, herbs, 2 tablespoons of the dressing. Taste and season with salt and pepper as needed.

Slice the chicken. Divide lettuce between 4 dinner plates or shallow bowls, Arrange sliced chicken, tomato/cucumber salad and potatoes over lettuce. Drizzle chicken with a little more of the dressing. Pass extras dressing at the table.