Bring a large pot of salted water to a boil over high heat. Cut the broccoli tops into bite-size florets; trim and peel the stems and slice them 1/2 inch thick (about 5-6 cups total). Add the broccoli. Cook for 3 minutes. Using a slotted spoon, transfer the broccoli to a bowl.
Add the angel hair to the boiling water and cook until al dente, following the package directions. Drain well and return the cooked pasta to the cooking pot.
In a separate non-stick pan, saute the mushrooms over medium high heat until soft and tender, for about 5 minutes. Lightly season with salt and pepper.
Meanwhile, combine the chicken stock and red pepper flakes in a small saucepan and bring to a boil over high heat.
Add the broccoli, chicken stock, and mushrooms to the drained pasta. Heat over medium heat until hot; toss together and add salt to taste. Add in the crumbled feta cheese and fresh parsley and toss together one last time. Serve and enjoy!