Crumble feta cheese and mix with chopped cilantro. Set aside.
Drizzle a little oil over the chicken breasts and rub all over. Season with salt and pepper on all sides.
Lift the skin on the chicken breast on one side and run a finger inside, making a pocket. Stuff about 2 tablespoons of sauteed mushrooms per chicken breast, followed by a tablespoon of feta cheese mix.
Place chicken breasts in a baking dish, skin side up, carefully pour in the broth, and bake for about 45 minutes, or until the internal temperature registers 160F. If you want a crispier skin, turn on the broiler on low during the final 5 minutes of baking.
Remove chicken breast from oven and let rest for 5 minutes before serving.