Position the oven racks such that you can fit a large, covered Dutch oven inside it (make sure it is at least 5.5 quarts, which will be a tight fit). Preheat to 325°F. Bring the lamb shanks to room temperature. Season all over with 2 teaspoons salt and 1 teaspoon pepper.
In a large Dutch oven or similar large, heavy-bottomed pot, heat 1 tablespoon of the canola oil over medium-high heat. Once the oil is hot and shimmering, add two of the lamb shanks. Brown on all sides, turning them every few minutes, until they are seared all over—this process will take about 8 minutes or so. Remove to a plate. Repeat with the following shanks (you shouldn't need more oil at this point, but if you do, go ahead and drizzle some in if the pot seems dry).
Add the remaining 1 tablespoon oil to the now empty pot and reduce the heat to medium. Add the onions, carrots, and celery. Cook until the vegetables begin to soften, 8 to 10 minutes.
Add the garlic, cumin, oregano, and fennel seed (if using), and cook for about 30 seconds, until the garlic is fragrant.
Pour in the wine, stirring to scrape up the stuck on bits from the bottom of the pan.
Add the crushed tomatoes, bay leaves, broth, and remaining 2 teaspoons salt and 1 teaspoon pepper.
Return the lamb shanks to the pot, nestling them into the liquid (the pot will likely be VERY full). Increase the heat and bring to a gentle simmer. Once the sauce is bubbling, cover the pot, then carefully transfer it to the oven.
Let the lamb shanks braise for 2 hours, uncovering and using tongs to carefully turn them once halfway through.
Uncover and continue baking for 30 minutes more. The lamb should be super tender and barely hanging onto the bone; if it's not, continue cooking uncovered longer as needed.
Remove the lamb to a serving plate and cover to keep warm. Fish out and discard the bay leaves.
Carefully strain the cooking liquid into a large bowl, pressing the veggies against the strainer to extract as many juices as possible. Discard the veggies; they've done their job!
Return the sauce to the pot and simmer a few minutes to thicken it further if desired. Serve the lamb shanks over the mashed potatoes, with the sauce spooned over the top and a sprinkle of parsley as desired.
Notes
TO STORE: Transfer leftover lamb shanks to an airtight container with the sauce. Refrigerate for up to 4 days.
TO REHEAT: Warm leftover lamb shanks with the sauce in an oven-safe skillet or covered in a baking dish. Reheat in the oven at 350ºF until it’s warmed through. The stovetop will also work, but you may need to add a splash of broth.
TO FREEZE: Transfer leftovers to a freezer bag and press out the excess air, or place the lamb shanks in an airtight container. Freeze for up to 3 months and thaw in the refrigerator before reheating.