Preheat your oven to 350 degrees. In a large pot of salted water, cook the jumbo pasta shells until al dente. Drain; let cool so you can handle them without burning your hands.
Meanwhile, heat the olive oil in a large skillet. Sauté the onion & asparagus over medium heat until tender. Stir in the chicken & minced garlic; cook for another couple minutes. Remove from the heat to cool.
In a large bowl, mix together the ricotta cheese, shredded cheese, egg, & some salt & pepper to taste. Stir in the chicken asparagus filling. Stuff the cheese mixture inside the cooked shells. Place them in a 2 quart baking dish.
Melt the butter in a large saucepan. Add the garlic, cream, & a pinch of pepper. Cook for about 5 minutes, whisking often. Add the Parmesan cheese and cook the sauce on medium heat for 10-15 minutes, or until the sauce has thickened and is smooth. Make sure you give it a good whisk every couple of minutes! Add the mozzarella cheese and stir until smooth.
Pour the Alfredo sauce over the filled shells. Bake at 350 for 30 minutes, or until the cheese sauce has browned slightly & is bubbly. Let cool for 5 minutes before serving.