Bring a large pot of well-salted water to a boil. Cook the pasta 1 minute shy of al dente. Reserve 1/2 cup of the pasta water, then drain.
Season the chicken with salt and pepper. Heat the olive oil in a large skillet over medium-high heat. Sear the chicken in a single layer until golden and cooked through (165°F internal, per USDA), 5 to 6 minutes. Transfer to a plate. (Don't crowd the pan — sear in two batches if needed. Crowding steams the chicken instead of browning it.)
Reduce the heat to medium. Add 2 tbsp butter to the same skillet. Add the garlic and cook 30 seconds until fragrant — no longer, or it turns bitter.
Pour in the heavy cream, scraping up the browned bits. Simmer gently 2 to 3 minutes until slightly thickened.
Off the heat, whisk in the remaining 2 tbsp butter, the Parmesan, Italian seasoning, and nutmeg until smooth. If the sauce is too thick, loosen it with a splash of the reserved pasta water.
Add the pasta and chicken back to the skillet. Toss over low heat for 1 minute so every noodle gets coated.
Shower with parsley and serve immediately with extra Parmesan.