In a large, heavy-bottomed Dutch oven or saucepan, heat 2 tablespoons of oil over medium-high heat. Once hot, add the beef short ribs and brown them for 3–4 minutes on each side until a deep golden crust forms.

In the same pot, lower the heat to medium and add roughly chopped onions, carrots, celery, and mushrooms. Sauté for 7–10 minutes until the vegetables soften and the onions become translucent.

Stir in the chopped garlic and red pepper flakes. Cook for an additional minute, allowing the garlic to become fragrant without burning.

Return the browned beef short ribs to the pot, nestling them into the vegetables. Pour in the beef broth and dry red wine, then stir in the tomato paste, whole garlic heads, balsamic vinegar, bay leaves, thyme, and rosemary.

Cover the pot and reduce the heat to a simmer, allowing the mixture to cook for 2–3 hours until the beef is tender.

Remove the bay leaves and adjust seasoning with salt and pepper to taste. Serve your French Style Braised Short Ribs over creamy mashed potatoes or with crusty bread.