Cook pasta according to package instructions, just until al dente. Drain from water and lay out on a large dish to cool, so that they don't stick together.
Make alfredo sauce:
Combine butter, garlic, heavy cream and milk in a saucepan over medium heat. Bring to a simmer.
Beat egg yolks in a small bowl. Add a spoonful of the hot milk mixture to the eggs, stirring to temper the eggs. Repeat with several more spoonfuls. Slowly add the egg mixture to the saucepan and stir to combine. Remove from heat.
Stir in Parmesan cheese and mozzarella. Season with salt and pepper, to taste.
In a large mixing bowl toss together chicken, broccoli, and 1/2 cup alfredo sauce.
Add a spoonful of alfredo sauce to the bottom of a 9x13'' pan. Fill shells with a large spoonful of chicken and broccoli mixture and place in prepared pan.
Pour remaining alfredo sauce evenly over the shells. Sprinkle with remaining 1/2 cup mozzarella cheese.
Bake at 350 degrees F for about 25 minutes or until warm and bubbly.
Make Ahead Instructions: Assemble stuffed shells through step 8 of the recipe, just before baking. Cover pan and refrigerate up to 1 day in advance, until ready to bake.Freezing Instructions: Cook the noodles 2-3 minutes less than al dente. Assemble the dish following the recipe directions but don't bake it (consider using a disposable container or aluminum pan). Cover with a layer of plastic wrap and then a double layer of aluminum foil and freeze for up to 2 months. Thaw overnight in the fridge. Remove the plastic wrap, cover it back up with aluminum foil, and bake for 45 minutes. Remove foil and bake uncovered for 15-20 more minutes or until warmed through and bubbly.