Bring a large pot of salted water to a boil. Add the penne pasta and cook according to package instructions until al dente. Drain and set aside.
While the pasta cooks, heat olive oil in a large skillet over medium-high heat.
Season the chicken pieces with 1/2 teaspoon salt and 1/4 teaspoon black pepper.
Add the chicken to the skillet and cook for 5-7 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, add the minced garlic and cook for 30 seconds until fragrant.
Add the fresh spinach and cook until wilted, about 2 minutes.
Pour in the chicken broth, lemon juice, lemon zest, remaining 1/2 teaspoon salt, 1/4 teaspoon black pepper, and crushed red pepper flakes. Stir to combine and bring to a simmer.
Return the cooked chicken to the skillet and add the drained pasta. Toss everything together until well combined and heated through, about 2 minutes.
Remove the skillet from heat and stir in the grated Parmesan cheese until melted and creamy.
Serve immediately, garnished with additional Parmesan if desired.