When you get right down to it, this is a simple dish of seared halibut, an elemental stew of guanciale and vegetables, wilted romaine lettuce, and parsnip puree.
Position a rack in the center of the oven and preheat the oven to 350° F.
Line a plate with paper towels. Heat a large skillet over low heat and add 1 tablespoon of the olive oil, warming it. Add the guanciale and cook, stirring occasionally with a wooden spoon, until the guanciale has rendered its fat and is browned and crisp, 10 to 12 minutes. Transfer the guanciale to the paper-towel-lined plate.
Drain all but 2 tablespoons of the fat from the pan and add the carrots, onions, and celery. Cook, stirring frequently, until the vegetables are softened but not browned, about 6 minutes. Remove the pan from the heat and set aside.
Heat a large heavy ovenproof skillet over high heat. Add 2 tablespoons of the olive oil and tip and tilt the pan to coat it, heating the oil until it is shimmering and almost smoking. Season the halibut with salt and pepper, add the fillets to the pan, skinned side down, and cook until golden brown, 3 to 4 minutes. Transfer the skillet to the oven and bake until the halibut is just cooked through, about 4 more minutes.
While the halibut is in the oven, return the guanciale to the pan with the vegetables, set over medium-high heat, season with salt and pepper, and toss to rewarm the guanciale and integrate the flavors. Transfer the mixture to a heat-proof bowl.
Add the remaining 1 tablespoon olive oil to the pan, add the romaine cups, and cook briefly, turning with tongs, to wilt them and flavor them with the lingering fat in the pan, about a minute. Remove from heat.
Spoon a line of parsnip puree down the center pf each of 4 dinner plates. Set a halibut fillet on one side of the puree and a romaine cup on the other. Spoon the guanciale mixture into the romaine cups, drizzle the fish with the lemon juice, and serve.