Season the lamb shanks all over with salt and pepper.
Heat the canola oil in a large Dutch oven over medium-high heat.
Sear 2 lamb shanks at a time until a dark brown crust forms on one side, 3-5 minutes. Sear the other sides of the meat until evenly browned. Remove the shanks from the pan and set aside. Remove all but 2 tablespoons of fat from the pan.
Add the minced carrots, onion, celery, and minced garlic to the pot. Season with salt and pepper, and cook until the vegetables are deeply caramelized, 15-20 minutes.
Add the tomato paste and cook until it browns and is aromatic, 5 minutes.
Add the red wine and scrape the bottom of the pot to release the flavorful browned bits stuck to the bottom. Cook until the wine reduces by half and thickens slightly, 3 minutes.
Add the chicken stock, orange juice, lemon juice, rosemary, thyme, and bay leaves. Stir and bring to a boil.
Return the lamb shanks to the Dutch oven, cover, and place in the oven for 2 hours, turning the shanks halfway through, until the lamb is tender and the braising liquid has thickened.
About halfway through the lamb cooking time, add the oblique-cut carrots, Brussels sprouts, red pearl onions, and whole garlic cloves to a baking sheet. Drizzle with olive oil and season with salt and pepper. Toss to combine. Roast in the oven for 1 hour, or until the vegetables are tender.
Right after the potatoes finish boiling, drain and transfer to a large bowl. Season with plenty of salt and a bit of white pepper, and mash the potatoes until as smooth as possible.
Fold in the butter, sour cream, and heavy cream until smooth and light.
Make the citrus gremolata: In a small bowl, combine the parsley, lemon zest, flaky salt, garlic, chives, horseradish, and orange zest.
To serve, scoop mashed potatoes onto a plate. Add roasted vegetables and a lamb shank. Top with the braising liquid and citrus gremolata.