Sear Chicken: Season both sides of the chicken with 3/4 teaspoon of salt and 1/4 teaspoon ground pepper. Heat pan over medium-high and add olive oil. Sear chicken 1-2 minutes on each side and then remove from the pan and set aside.
Saute Onions: Deglaze the pan with red wine vinegar and another tablespoon of olive oil, use a wooden spoon to scrape up the browned bits off of the bottom of the pan. Turn heat down to medium, and add onions, garlic, and shallots and saute for 5 minutes.
Add Stock and Veggies: Add chicken stock to the pan and then return chicken to the pan and add the green beans and mushrooms to the pan. Season veggies with the last teaspoon of salt and the parsley and thyme. Depending on the size of the pan you use, this will look like it's overflowing, don't worry, it all cooks down. The most important thing is that the chicken is in the stock to braise.
Cook: Place skillet in the oven uncovered to braise. Cook for 25 minutes and stir veggies so that they all get coated with stock. Return to oven to finish cooking for the final 20 minutes until chicken is fully cooked and the internal temperature reaches at least 165 degrees internal temp.
Notes
Pan SizeUse at least a 3.5 quart pan for 1/2 lb of each green beans and mushrooms. If you have a larger pan, up to 6 quart, then you can increase the green beans and mushrooms.