Boil the pasta. Bring a large pot of salted water to a rolling boil.
Add pasta and cook until just shy of al dente according to package directions. Scoop out about 1/2 cup of pasta water and set aside. Drain the pasta.
Warm the chicken. While the pasta cooks, pull the chicken from the rotisserie and shred or chop it into bite-size pieces.
Aim for about 2 cups.
Sauté the garlic. In a large skillet over medium heat, melt the butter with the olive oil. Add the minced garlic and cook 30–60 seconds, stirring, until fragrant. Don’t let it brown.
Build the sauce. Pour in the heavy cream and milk.
Stir and reduce heat to medium-low. Let it gently simmer for 2–3 minutes to warm through and slightly thicken.
Add cheese and season. Stir in the Parmesan a handful at a time until melted and smooth. Season with salt, black pepper, red pepper flakes, and nutmeg if using.
If the sauce seems too thick, splash in a bit of the reserved pasta water or more milk.
Combine pasta and chicken. Add the cooked pasta and shredded chicken to the skillet. Toss to coat, adding a little pasta water as needed to loosen. The sauce should cling to the noodles without pooling.
Finish and rest. Cook together for 1–2 minutes on low, just until everything is hot and glossy.
Remove from heat and let it sit for 1 minute to set.
Garnish and serve. Top with extra Parmesan and chopped parsley. Taste and adjust salt and pepper. Serve warm.