Heat grill to medium-high. Once it is hot, clean the grill grate with a wire brush. Just before grilling, oil the grill grate.
Whisk together the lemon juice, rosemary, garlic, 3 tablespoons of the olive oil, ½ teaspoon salt, and ¼ teaspoon pepper in a medium bowl.
Transfer half the dressing to a small bowl; set aside. Add the chicken to the remaining dressing and turn to coat.
Rub the corn with the remaining tablespoon of oil and season with ¼ teaspoon each salt and pepper.
Grill the corn and chicken, covered, turning occasionally, until the corn is tender and lightly charred, 4 to 6 minutes, and an instant-read thermometer inserted in each pieces of chicken registers 165º F, 8 to 10 minutes.
Cut the kernels off the cobs and slice the chicken.
Toss the spinach, chicken, corn, and avocado with the reserved dressing and sprinkle with the Parmesan.