Bring a large pot of salted water to a boil. Add the penne pasta and cook according to package instructions until al dente, about 10-11 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 1/2 teaspoon kosher salt and 1/4 teaspoon black pepper.
Add the chicken to the skillet and cook until golden brown and cooked through, about 6-7 minutes, stirring occasionally. 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 and bring to a simmer.
Return the cooked chicken to the skillet and stir to combine.
Add the cooked pasta to the skillet and toss everything together.
Stir in the grated Parmesan cheese, lemon juice, lemon zest, remaining 1/2 teaspoon kosher salt, and 1/4 teaspoon black pepper.
Cook for an additional 2 minutes, stirring frequently, until the cheese melts and the sauce coats the pasta evenly.
Remove from heat and serve immediately.