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.

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 ½ cup alfredo sauce.

Add a spoonful of alfredo sauce to the bottom of a 9×13'' 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 ½ cup mozzarella cheese.

Bake at 350 degrees F for about 25 minutes or until warm and bubbly.