Bring a large pot of salted water to a boil and cook the pasta according to package directions until al dente; reserve 1 tablespoon pasta cooking water if desired, then drain and set the pasta aside.
Return the same large pot to the stove over medium heat and melt 1 tablespoon butter.
Add the chopped garlic or shallots and sauté until fragrant and tender, about 1 minute.
Pour in 1/2 cup heavy cream and add 1/2 cup grated Parmesan; stir until the cheese begins to melt and the sauce is smooth.
Add the drained pasta and 2 cups cooked chicken to the sauce, tossing to combine; if the sauce seems too thick, stir in up to 1 tablespoon reserved pasta water.
Season to taste with salt (and pepper if desired), then sprinkle with chopped chives and top with mozzarella if using.
If you want melted, browned mozzarella, place the pot under a preheated broiler for 1–2 minutes until the cheese is bubbly and lightly golden, watching carefully.