Bring a large pot of salted water to a boil. Cook the penne pasta according to package instructions until al dente. Drain and set aside.
Heat olive oil in a large skillet over medium-high heat. Season the chicken pieces with salt and black pepper.
Add the chicken to the skillet and cook for 5 to 6 minutes, stirring occasionally, until the chicken is cooked through and lightly browned. Remove the chicken from the skillet and set aside.
In the same skillet, reduce heat to medium and add the minced garlic and red pepper flakes. Sauté for about 1 minute until fragrant, stirring constantly to avoid burning.
Pour in the chicken broth and water, scraping the bottom of the skillet to loosen any browned bits. Bring the liquid to a simmer.
Add the cooked pasta to the skillet and stir to combine. Let it simmer for 2 to 3 minutes so the pasta absorbs some of the broth.
Add the fresh baby spinach to the skillet and cook for 1 to 2 minutes until wilted.
Return the cooked chicken to the skillet and stir to combine all ingredients.
Remove the skillet from heat. Stir in the grated Parmesan cheese, lemon juice, lemon zest, and unsalted butter until the cheese melts and the sauce becomes creamy and well combined.
Taste and adjust seasoning with additional salt and pepper if needed.
Serve the chicken spinach skillet pasta immediately, garnished with extra Parmesan if desired.