1-1/2 lb. baby potatoes, scrubbed, halved or quartered if larger than 1-1/2 inches
2/3 cup extra-virgin olive oil
1-1/4 tsp. finely grated zest, plus 3 Tbs. fresh lemon juice (from 1 large lemon)
1 large clove garlic, mashed to a paste with 1/2 tsp. kosher salt
Freshly ground black pepper
4 small boneless, skinless chicken breast halves (about 1-1/2 lb.)
1/2 large seedless cucumber, peeled in stripes, seed core removed, cut into 1/2-inch pieces
1 pint cherry tomatoes, halved
1/4 cup coarsely chopped fresh mixed herbs, such as basil, cilantro, chives, tarragon, parsley, and mint (use at least 3)
Prepare a medium-high gas or charcoal grill fire.
Put the potatoes in a 3-quart saucepan of well-salted water. Boil over high heat until nearly cooked through but still a little firm, about 5 minutes.
Shake the oil, lemon zest and juice, garlic paste, 1/2 tsp. salt, and 1/4 tsp. pepper in a jar with a lid.
Drain the potatoes, transfer to a large bowl, and toss with 3 Tbs. of the vinaigrette. Toss the chicken in a medium bowl with 3 Tbs. of the vinaigrette, 1/2 tsp. salt, and 1/8 tsp. pepper. Arrange the chicken on one side of the grill grate. Put a large piece of foil on the other side of the grill and transfer the potatoes to the foil with a slotted spoon. Cover the grill and cook, turning the chicken and potatoes after about 5 minutes, until the chicken is just cooked through and the potatoes are browned, 10 to 12 minutes total. Let the chicken rest 5 minutes before slicing.
Meanwhile, in a medium bowl, combine the cucumber, tomatoes, herbs, 2 Tbs. of the vinaigrette, and salt and pepper to taste. Slice the chicken, arrange on dinner plates, and drizzle with the remaining vinaigrette. Serve with the potatoes and the cucumber salad.