To cook the dish, heat a large pan over medium-high heat. Once hot, add about 1 tablespoon of the oil. Add the chicken, spreading it out slightly, so that it cooks evenly. Season with salt and pepper and cook just until barely cooked through (the chicken will be added back to the sauce later to finish cooking). Remove it from the pan and set aside, while you cook the mushrooms.
Add a touch more oil and sauté the mushrooms until they start to release their juices and begin to brown. Then add the sun-dried tomatoes and chili flakes, followed by the garlic. Sauté everything for about 30 seconds and then deglaze with the white wine. Let the wine reduce down a bit, before adding the chicken stock. Bring the sauce to a boil, and then lower the heat. Let the sauce simmer until it reduces by about a third.
While the sauce reduces, cook the pasta in well-seasoned, rapidly boiling water. About a minute before the pasta is ready, add the chicken to the sauce to cook through. Once the pasta is al dente, reserve about one cup of the cooking water, just in case you need to thin out the sauce.
Drain the pasta, add it directly to the skillet and gently toss to combine. Here is where you would add a touch of the cooking liquid, if the pasta looks a bit dry.
To finish, break up the feta cheese into big pieces. Add the feta and parsley, and gently toss everything together. Taste for seasoning and serve immediately with parmesan cheese and a drizzle of extra-virgin olive oil, if desired.