Dice chicken, mix with oil and seasonings, refrigerate 30 minutes.
Cook chicken in butter and oil 6–7 minutes until golden. Set aside.
Sauté chopped broccoli in the same skillet about 3 minutes.
Melt butter, add garlic, cream, cream cheese, and Parmesan. Stir until smooth and thick.
Boil jumbo shells in salted water until al dente. Reserve ½ cup pasta water, then drain.
Mix chicken, broccoli, mozzarella, Parmesan, and half the sauce. Add pasta water if needed.
Preheat oven to 375°F. Spread sauce in baking dish and fill shells with the mixture.
Pour remaining sauce over shells, add mozzarella, bake 15 minutes.
Rest 5 minutes, then serve warm.