Preheat a grill to medium. Combine the parsley, 3 tablespoons olive oil, 1 tablespoon water, the garlic, shallot, lemon zest and juice, and 1/2 teaspoon salt in a mini food processor and puree.
Toss the squash with 2 tablespoons of the parsley sauce; spread in a single layer in a large cast-iron skillet. Transfer the skillet to one side of the grill; cook until the squash softens, about 10 minutes.
Meanwhile, rub the chicken with the remaining 1 tablespoon olive oil and season with salt and pepper. Grill on the other side of the grill until just cooked through, 4 to 5 minutes per side. Top each chicken breast with a slice of cheese, then a slice of ham. Cover and cook until the cheese melts, about 2 more minutes.
Serve the chicken with the salad greens and squash. Top with the remaining parsley sauce.