Bring a large pot of salted water to a boil for pasta. Meanwhile, heat grill to medium; lightly oil grates. Season chicken with salt and pepper. Grill until cooked through, 2 to 3 minutes per side. Remove from grill, and cut into thin strips, halving if long.
Cook pasta until al dente, according to package instructions. Reserve 1 cup pasta water; drain pasta, and return to pot. Add chicken, tomatoes, mozzarella, parsley, Parmesan, and butter. Toss to combine. Add reserved pasta water a little at a time to create a sauce that coats shells (you may not need all the water). Serve with more Parmesan.
MAKE-AHEAD IDEA If youâ€™d like, grill the chicken up to two days beforehand. Then let it cool, wrap it in plastic, and refrigerate. Bring chicken to room temperature before adding to pasta.