Bring a large pot of salted water to a boil and cook the jumbo pasta shells according to package directions until they're al dente (firm but not hard). Drain the shells in a colander and spread them out on a sheet pan or clean kitchen towel to cool and prevent sticking. While the pasta cooks, mince the garlic cloves, chop the broccoli into small florets, and ensure your chicken is shredded into bite-sized pieces. Have all ingredients measured and ready before you start the sauce.

Melt the butter in a large saucepan over medium heat, then add the minced garlic and cook for about 1 minute until fragrant—don't let it brown. Pour in the cream and milk, stirring to combine, and bring the mixture to a gentle simmer. Let it simmer for 2-3 minutes to allow the flavors to meld together.

In a small bowl, whisk the egg yolks together. Slowly add about 1/2 cup of the hot cream mixture to the egg yolks while whisking constantly—this tempering process prevents the eggs from scrambling and creates a silky sauce. Once the eggs are warmed, remove the saucepan from heat and slowly pour the egg mixture back into the pan while stirring constantly. I find that taking this step slowly and deliberately ensures you get that extra creamy, rich texture without any scrambled bits.

Stir in the parmesan cheese, the 1/4 cup of mozzarella, a pinch of nutmeg, and season with salt and pepper to taste. The sauce will be silky and smooth. Set aside about 1/2 cup of this sauce in a separate bowl for mixing with the filling; you'll use the remaining sauce for layering in the baking dish.

In a medium bowl, combine the shredded chicken, chopped broccoli, and the reserved 1/2 cup of sauce from Step 4, mixing until everything is well coated. Spread a thin layer of the remaining Alfredo sauce on the bottom of a 9x13 inch baking dish. Fill each cooled pasta shell with about 2-3 tablespoons of the chicken-broccoli filling and arrange them seam-side up in the baking dish. I like to arrange them snugly so they support each other and hold their filling during baking.

Pour the remaining Alfredo sauce from Step 4 evenly over all the filled shells, making sure each one gets a good coating. Sprinkle the 3/4 cup of mozzarella cheese over the top. Bake in a preheated 350°F oven for 25-30 minutes until the shells are heated through and the cheese is melted and lightly golden on top. Let the dish rest for 5 minutes before serving—this brief rest helps the sauce set slightly and makes plating much easier.