Bring large pot of salted water to boil for pasta.
Meanwhile, heat grill to medium, lightly brush with oil.
Season chicken with salt and pepper.
Grill until cooked through, 2 to 3 minutes per side.
Remove from grill and gut into thin strips.
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.
Serve with more parmesan.