Cook pasta in a large pot of boiling water with 1 teaspoon of salt added to it. Cook more shells than needed, as some of them may break while cooking.
In a large saucepan or pot, heat 1 jar of alfredo sauce until it is simmering. Season with pepper.
Stir in parmesan and mozzarella cheese and heat until melted and combined.
Stir in chicken and broccoli and heat over low until warmed through.
Spoon filling into the cooked pasta shells.
Add warmed alfredo sauce to the bottom of the baking dish. Arrange stuffed shells over the alfredo sauce in the dish. Drizzle additional alfredo sauce on top.
Cover the baking dish with aluminum foil and bake for 25-30 minutes until the shells are heated through and the sauce is bubbly.
Notes
You could also place the filled shells in a baking dish and refrigerated so they can be heated through at a later time. Another option is to sprinkle additional mozzarella cheese over the top of the shells and bake.
Use a pastry bag without a tip or a Ziploc bag with the end cut off to fill your stuffed shells easily. If the sauce is too hot to handle, try wrapping the pastry bag with a tea towel.
This dish can be made ahead of time and refrigerated until ready to bake. Just add about 15 minutes to the bake time.