French Style Braised Short Ribs

Name: French Style Braised Short Ribs
Ingredients: 2 pounds beef short ribs, cut into 3-4 inch pieces; 1 large onion, sliced; 3 carrots, sliced; 3 stalks celery, sliced; 8 ounces mushrooms, quartered; 4 cloves garlic, chopped; 1 pinch red pepper flakes (optional); 3 cups beef broth; 2 cups dry red wine (or more broth); 2 tablespoons tomato paste; 2 heads of garlic, with the tops removed; 2 tablespoons balsamic vinegar; 2 bay leaves; 1 teaspoon thyme, chopped (or 1/2 teaspoon dried thyme); 1 teaspoon rosemary, chopped (or 1/2 teaspoon dried rosemary); 2 tablespoons parsley, chopped