Cook the pasta. Bring a large pot of salted water to a boil. Then, add the penne pasta and cook until al dente, according to package instructions. After it's cooked, drain the pasta, toss it with olive oil to prevent sticking, and set aside while you prepare the sauce and chicken.
Cook the chicken. Heat the olive oil in a deep 12-inch skillet over medium heat until it shimmers. Then, add the chopped chicken pieces in one layer, and season with salt, black pepper, and crushed red pepper. Cook for 5-6 minutes, flipping the chicken pieces halfway through, until they are golden brown and cooked through. Remove the chicken from the skillet and set it aside.
Make the sauce. In the same skillet, add the minced garlic, and stir frequently and cook for 1-2 minutes until the garlic is fragrant but not burned. Next, pour in the chicken broth or white wine, stirring to deglaze the pan and lift any browned bits from the bottom. Then, reduce the heat to low, then stir in the heavy cream and grated Parmesan cheese. Simmer gently for 1-2 minutes until the cheese melts and the sauce is smooth. Add the cooked chicken back into the skillet and let it simmer in the sauce for about 5 minutes until the sauce thickens slightly.
Add in the spinach and pasta. Remove the skillet from the heat. Then, stir in the fresh spinach, folding it gently into the hot sauce and chicken until it wilts. Next, add the cooked pasta and toss everything together until well combined and the pasta is evenly coated with the creamy sauce.
Serve. Top with extra grated Parmesan cheese, if desired, and serve immediately.