Preheat the oven to 400°F.
Clean the artichokes: Fill a large bowl with cold water, then halve, squeeze, and drop two of the lemons into the water. Peel away and discard the outer artichoke leaves until you reach the tender, pale green leaves. Using a vegetable peeler, peel the stem, removing any small tough leaves that remain. Using a chef’s knife, remove the top 2–3 inches of the artichoke, then use a spoon to scoop out the fibrous inner “choke.” Cut the artichoke hearts into quarters, and drop them in the lemon water while you continue cleaning the remaining artichokes.
Drain the artichoke pieces, and season them lightly with salt and pepper. To a large Dutch oven over medium-low heat, add ¼ cup olive oil. When the oil is just beginning to shimmer, add the artichoke hearts to the skillet, cut side down, and cook, turning occasionally, until lightly browned all over, 5–8 minutes. Use tongs or a slotted spoon to transfer the artichokes to a heatproof plate and set aside. Add the onions, carrots, and garlic to the skillet, lower the heat to medium-low, cover, and cook, stirring occasionally, until the vegetables just begin to soften, 10–12 minutes. Remove the lid, return the reserved artichoke hearts to the skillet, then add the stock, wine, and the bouquet garni. Season lightly with salt, bring to a simmer, and cook until the vegetables are just tender when pierced with a knife, 35–40 minutes. Add the anchovy and continue cooking 5 minutes more. Season to taste with additional salt and black pepper, remove from the heat, cover, and keep warm while you cook the fish.
Use paper towels to pat the halibut filets dry, then season lightly with salt and pepper. To a large cast iron skillet over medium-high heat, add the canola oil. When the oil is shimmering and beginning to smoke, add the fish.Turn the heat to medium and cook without disturbing until the halibut begins brown on the bottom, about 4 minutes. Transfer the skillet to the oven and cook until the fish is just barely opaque, 4–6 minutes. Remove the pan from the oven and place over medium heat. Add the butter to the pan to melt and, using a spoon, quickly baste the fish with the buttery cooking juices. Use an offset spatula to flip the fish so the browned side faces up. Squeeze the remaining lemon halves over the fish.
Distribute the reserved vegetables and their cooking liquid among 4 wide bowls. Top each with a piece of halibut and drizzle with additional olive oil. Garnish with parsley and flaky sea salt and serve hot.