Bring a large pot of salted water to a boil. Cook the tortellini according to package directions. 2 minutes before the tortellini finishes cooking, add the broccoli to the pot.
As the tortellini cooks, prepare the chicken and alfredo sauce. Melt half of the butter in a large skillet over medium-high heat. Season the chicken generously with salt and pepper. Add the chicken to the skillet and cook until well-browned all over and cooked through, about 5-7 minutes. Remove the chicken from the skillet and set aside.
Add the garlic to the skillet and cook until fragrant, about 30 seconds. Pour in the heavy cream and place the remaining butter in the skillet. Bring to a simmer, whisking constantly. Whisk in the parmesan cheese, until fully melted and thickened. Whisk in half of the mozzarella cheese.
Place the chicken, tortellini, and broccoli in a 2.7 quart baking dish. Pour over the Alfredo sauce and toss to combine. Top with the remaining mozzarella.
Bake 10 minutes, or until the cheese begins to bubble and brown. Garnish with chopped fresh parsley.