Bring a large pot of salted water to a boil and cook the penne until al dente according to package directions; reserve about 1/2 cup pasta water, then drain the pasta.
While the pasta cooks, heat a large skillet or braiser over medium heat and add the reserved sun-dried tomato oil (about 3 tablespoons).
Add the cubed chicken to the skillet along with the chopped sun-dried tomatoes, 1/2 teaspoon salt, 1/2 teaspoon pepper and 1/2 teaspoon red pepper flakes; cook, stirring occasionally, until the chicken is cooked through and no longer pink, about 5 minutes.
Stir in the chopped tomatoes and minced garlic and cook for 2–3 minutes until the tomatoes begin to soften.
Add the fresh spinach and cook, stirring, until it wilts down, about 1–2 minutes.
Add the drained pasta to the skillet, sprinkle in 1/2 cup of the grated Parmesan, and toss to combine; add reserved pasta water a little at a time if the mixture seems dry to reach desired creaminess.
Adjust seasoning to taste, drizzle with a little extra olive oil if desired, sprinkle with the remaining Parmesan, and serve immediately.