Preheat oven to 375°F. Season chicken with 1 t salt and ½ t of black pepper. Heat 2 T oil in large skillet over medium-high heat, add chicken and cook until browned on both sides, 6-10 minutes total. Set aside.
Add 2 T oil to pan along with onions and mushrooms. Reduce heat to medium and cook, stirring often, until softened, about 3 minutes. Stir in garlic and cook about 30 seconds. Spread onion mixture evenly over bottom of a 9x13-in baking dish.
Pour ¼ c chicken broth into pan and scrape up any browned bits on the bottom. Reduce broth by half and then pour in remaining broth. Turn off heat and whisk in Chobani and the remaining 1½ t salt.
Evenly layer rice over onion mixture and then pour broth over rice. Sprinkle mixed fresh herbs over top and add chicken in a single layer on top.
Cover pan tightly with foil and bake for 1 hour and 15 minutes. Remove from oven and let stand 10 minutes before removing foil. Sprinkle with parsley before serving.