In a large pot of salted boiling water, cook the pasta according to package instructions until al dente. Reserve 1 cup of pasta water, then drain and set aside.
In a large skillet, melt 2 tablespoons of butter over medium heat. Add the sliced chicken, season with salt and pepper, and cook until golden brown and cooked through, about 5-7 minutes. Remove the chicken from the skillet and set aside.
In the same skillet, add the remaining 2 tablespoons of butter. Add the minced garlic and sauté for about 1 minute or until fragrant.
Pour in the heavy cream, stirring constantly, and bring to a gentle simmer. Gradually whisk in the grated Parmesan cheese until smooth and creamy. If the sauce is too thick, add reserved pasta water a little at a time until desired consistency is reached.
Add the cooked chicken back into the skillet, along with the cooked pasta and baby spinach (if using). Toss everything together until well coated in the sauce. Adjust seasoning with salt, pepper, and red pepper flakes for a little heat, if desired.
Plate the pasta and garnish with chopped parsley. Serve immediately with additional Parmesan on the side, if desired.