Defrost, rinse and dry lamb shanks.-Liberally dust with flour, salt, and pepper.
Heat oil in Dutch oven till shimmering. Do not burn.
Brown shanks on all sides in hot oil, turning as the meat becomes deep brown. (Important for richer flavor to not skip this step.) Remove browned shanks to plate.
Add chopped vegetables to the pan drippings and sauté till they start to soften.
Add garlic, thyme, and tomato paste and stir till combined with vegetables.
Pour 1 cup wine (or chicken broth), and remaining broth into the vegetable mixture. Bring to a boil, then reduce to a simmer.
Add bay leaf and season to taste with salt and pepper. (I go light on the pepper and let folks add more to their plate when served.)
Return shanks to the Dutch oven and nestle them down into the vegetables and broth. Tightly cover and turn heat down to low simmer.
Braise, turning every 20 minutes or so, for 2 hours or until the meat is falling-off-the-bone tender. You may also move Dutch oven to your 300° oven to finish braising.