Prepare the roasted garlic and mushroom essence if you don’t already have these on hand.
Bring a pot of salted water to a boil. Have ready a bowl of ice water. Clean the crosnes, and cook them until just tender, about 6 minutes. Submerge in the ice water to cool. Drain, pat dry and set aside.
Place the sunchokes, garlic, thyme, parsley and salt in a heavy-bottomed pot. Pour in enough cold water to cover by 2 inches (5 cm), and bring to a boil.
Lower the heat to a gentle simmer, and cook until just tender. Turn off the heat, and allow the sunchokes to cool in the pot. Remove from the pot, discarding the cooking liquid. Cut the sunchokes in half lengthwise, and set aside.
Place the potatoes in a pot with a good pinch of salt and enough water to cover by 2 inches (5 cm). Bring to a boil, then lower the heat to a gentle simmer. Cook until tender and easily pierced with the tip of
a knife. Drain off the water and return the potatoes to a low heat. Cook slowly for about 10 minutes, allowing excess moisture to evaporate and shaking the pot occasionally (do not stir).
Meanwhile, in a small saucepan, bring the cream to a boil and remove from the heat.
Pass the potatoes through a food mill or potato ricer or push them through a coarse sieve into a clean frying pan or sauté pan. Put the pan over low heat, and beat in the 1 cup (250 mL) butter and the crushed roasted garlic. Beat in the cream a little at a time. Transfer to a bowl and keep warm. Clean the pan.
Preheat the oven to 425°f (220°c).
Season the fish with salt. Heat a large ovenproof non- stick frying pan, and add 2 Tbsp (30 mL) of the canola oil. Place the fish in the pan, skin side down. Cook until the skin starts to caramelize, then move to the oven for 6 minutes. Flip the fish over and drizzle with a little clarified butter. Return the pan to the oven for a further 3 minutes. Remove from the oven, and let stand uncovered in a warm place for 5 minutes, allowing the fish to cook in the residual heat.
Meanwhile, remove the outer leaves from the Brussels sprouts and discard. Separate the remaining leaves and set them aside. Heat a frying pan with 1 Tbsp (15 mL) of the canola oil. When the oil smokes, add the leaves, which should crackle and colour. Add a pinch of salt and the 1 tsp (5 mL) of unsalted butter.
Heat 2 frying pans, and add 2 Tbsp (30 mL) of canola oil to each.
In 1 pan, lightly sauté the mushrooms, finishing with 1 Tbsp (15 mL) of butter, the chives and a grind of pepper.
In the other pan, fry the sunchokes until golden brown and crispy, finishing with 1 Tbsp (15 mL) of butter. Season with salt.
Warm the mushroom essence.
To serve, spread some of the potato purée over one side of each plate. Spoon some of the crosnes and sunchokes over the other side. Place the fish on top of the vegetables, and sprinkle the mushrooms and Brussels sprouts over the plate. Finally, drizzle with mushroom essence and a small amount of olive oil.