Bring a large pot of salted water to a boil. Add the penne pasta and cook according to package instructions until al dente, about 10 minutes. Drain and set aside.
While the pasta cooks, heat olive oil in a large skillet over medium-high heat.
Season the chicken pieces with kosher salt and black pepper.
Add the chicken to the skillet and cook for 5-6 minutes, stirring occasionally, until the chicken is cooked through and lightly browned.
Add the minced garlic to the skillet and sauté for 1 minute until fragrant.
Pour in the chicken broth, scraping up any browned bits from the bottom of the skillet.
Stir in the cooked pasta, grated Parmesan cheese, lemon juice, lemon zest, and crushed red pepper flakes.
Add the fresh baby spinach and cook, stirring frequently, until the spinach is wilted, about 2-3 minutes.
Adjust seasoning with additional salt and pepper if needed.
Serve hot, garnished with extra Parmesan cheese if desired.