Preheat a grill to medium high. Lay out four 18-inch-long sheets of heavy-duty foil on a work surface. Generously season the chicken with salt and pepper. Toss the potatoes, lemon juice, 3 tablespoons olive oil, 2 teaspoons salt and 1/4 teaspoon pepper in a large bowl.
Divide the potatoes and any juices among the foil sheets, mounding them in the middle. Top evenly with the shallots, thyme and lemon wedges. Top each mound with a chicken breast. Brush the chicken with the remaining 1 tablespoon olive oil. Bring the short ends of the foil together and fold to seal, then fold in the sides.
Grill the foil packets seam-side up, 8 minutes. Flip the foil packet, gently press down and continue grilling until the chicken is cooked through and the potatoes are tender, about 6 more minutes.
When you include a lemon wedge in your foil packet, it gets extra juicy!