Season the lamb shanks with the salt and pepper. In a large dutch oven, or other heavy bottomed pot, over medium high heat, add 3 Tablespoons olive oil. Brown the shanks by cooking them on one side for 7 min, flipping and cook for them another 7 min on the other side.

Remove the lamb shanks from pot and set them aside.

Add the carrots, onions, celery to the pot you just browned the lamb in. Cook for 6 minutes, stirring frequently to break up any pieces remaining of lamb from the browning and so that the vegetables do not stick. Add the garlic and cook for another 2 minutes.

Add the red wine and broth, bay leaves, thyme, rosemary, tomatoes, tomato paste, stir to combine into the pot. Carefully lower shanks back into pot. The liquid should either cover the shanks entirely or cover them almost entirely. If you need more liquid, add equal parts chicken broth and red wine until they are covered.

Bring everything to a boil, reduce the heat to a simmer, cover, and simmer for 2.5 hours. To cook in the oven, cover the pot and bake at 350 F for 2.5 hours.

After cooking, remove shanks from pot. Strain the vegetables out of the sauce, discard the vegetabls, and return the sauce to pan. Crank the heat back up to medium high and cook uncovered to reduce the red wine sauce until it thickens to your liking, stirring occasionally. For me this was about 30 minutes but I'd start checking around 15 minutes.

Serve over mashed potatoes, rice, or pasta, with a generous helping of sauce poured over.