1. Poach the dried figs: Place port in a small saucepan over medium heat; reduce to 1 cup. Add star anise, thyme, and black peppercorns; remove from heat and let stand 20 minutes to infuse. Strain into a medium bowl; discard spices. Add figs to port reduction and let stand until tender, about 30 minutes.

2. Heat chicken stock in a medium skillet over medium heat. Drain figs and add to skillet along with butter and cook until figs are glazed, 1 to 2 minutes.

3. Roast the figs: Preheat oven to 350 degrees. Place sugar in a shallow dish. Press cut sides of figs into sugar. Heat canola oil in a medium ovenproof skillet over high heat. Using a paper towel, blot any excess oil from skillet. Place figs in skillet, cut-side down; cook until sugar caramelizes, about 1 minute. Turn figs and transfer skillet to oven. Roast for 1 minute.

4. Roast the white mushrooms: Heat oil in a skillet over medium-high heat. Place mushrooms in skillet, cut-side down, and cook for 1 minute. Add butter, thyme, and salt to skillet; cook until butter is melted. Continue cooking, basting mushrooms with melted butter, until mushrooms are tender, 1 to 2 minutes.

5. Make the fig jus: Heat canola oil in a large, straight-sided skillet over high heat. Add bones and sear until golden on all sides, 1 to 2 minutes. Remove bones from pan and set aside. Drain excess oil from skillet and discard.

6. Place skillet over medium heat; add figs, star anise, shallots, and guanciale; cook, stirring, until shallots are translucent and fat is rendered from guanciale, 4 to 5 minutes. Add vinegar to skillet to deglaze. Return bones to skillet along with chicken jus; bring to a simmer. Let simmer for 20 minutes. Strain and season with salt.

7. Make the loup de mer: Heat oil in a large skillet over medium-high heat until hot. Season fish with salt and place skin-side down in skillet, making sure skillet is hot or fish will stick. Reduce heat to medium-low and press fillets down with a metal spatula to keep flat as they begin to curl; let cook until skin is crisp, 3 to 4 minutes. Turn fillets and cook until flesh side is just warmed, 2 to 3 seconds. Transfer fish to a rack, skin-side up.

8. Serve: Drizzle raw mushroom slices with olive oil; set aside. Spoon 2 tablespoons sweet potato puree onto a plate and pull down center with a small spoon. Place 2 fish fillets in center of plate. Arrange 4 roasted fig halves, 2 glazed poached dried figs, 4 pickled porcini mushroom slices, 4 roasted porcini mushroom halves, 2 raw mushroom slices, and 2 slices Iberico ham around fish. Finish plate with fig jus and garnish with 4 lamb's-quarter leaves. Repeat three more times with remaining ingredients.