Preparation: Preheat your oven to 350°F (180°C) if choosing to oven-braise. In a large, oven-safe saucepan, sear the beef ribs over medium-high heat until browned on all sides, about 3-4 minutes per side. Remove ribs and set aside. Drain excess grease, leaving about 2 tablespoons in the pan.

Vegetables: Add onions, carrots, celery, and mushrooms to the pan. Sauté until the vegetables are tender, approximately 7-10 minutes.

Aromatics: Introduce the chopped garlic and red pepper flakes to the pan, cooking until fragrant, roughly 1 minute.

Braising Liquid: Pour in the beef broth and Worcestershire sauce. Stir in the tomato paste. Add whole heads of garlic, balsamic vinegar, bay leaves, thyme, and rosemary. Return the ribs to the pan, ensuring they are submerged.

Braise: Bring the mixture to a boil, then reduce heat to low, cover, and simmer for 2-3 hours. Alternatively, cover the pan and place it in the preheated oven, or transfer everything to a slow cooker. Cook until the beef is tender and falling off the bone.

Final Touches: Before serving, discard the bay leaves. Adjust seasoning with salt and pepper as needed. Serve the ribs over a bed of mashed potatoes, celery root puree, parsnips, or pasta. Accompany with crusty bread for dipping into the rich sauce. Garnish with fresh parsley and squeezed roasted garlic cloves from the heads.