Bring a large pot of salted water to a boil and cook the penne until al dente. Stir occasionally to prevent sticking. Drain the pasta and set it aside, reserving a little pasta water just in case the sauce needs thinning.
Heat olive oil in a large skillet over medium heat. Season the chicken pieces with salt and pepper, then add them to the pan in a single layer. Cook until golden on the outside and fully cooked through, then remove and set aside.
In the same skillet, lower the heat slightly and add the minced garlic. Stir constantly for about 30 seconds until fragrant. Be careful not to let it brown, as burnt garlic can taste bitter.
Pour in the chicken broth and scrape up any browned bits from the pan. Add the heavy cream and Italian seasoning, then let the mixture simmer gently. This step allows the sauce to thicken and develop flavor.
Stir in the fresh spinach and let it wilt naturally into the sauce. Add the grated Parmesan cheese slowly, stirring until fully melted and smooth. The sauce should be creamy and glossy.
Return the cooked chicken to the skillet and add the drained pasta. Toss everything together until the pasta is evenly coated with sauce. Adjust seasoning with salt, pepper, or red pepper flakes if needed.