Instructions

1. Using a paring knife, cut 1⁄2-inch off each tomato top and set aside. Using a melon baller or small spoon, scrape the insides from each tomato into a bowl, being careful not to tear through the skin. Sprinkle the insides of the tomatoes with salt, then invert onto a rack set over a baking sheet and let drain for 1 hour.
2. Meanwhile, heat the oven to 300°. In a medium Dutch oven, warm 1⁄4 cup duck fat over medium-high heat. Season the lamb shoulder with salt and pepper, then add to the pot and cook, turning, until golden brown on all sides, about 12 minutes.
3. Transfer the pot to the oven and cook, turning the meat once halfway through, until tender and falling off the bone, about 2 hours. Transfer the lamb to a rack and let cool. Shred the meat from the bone and transfer to a large bowl, discarding the bone.
4. Warm a large nonstick skillet over high heat. Add half the foie gras and cook, undisturbed, until dark brown on one side, about 30 seconds. Shake the pan to ensure that the foie gras doesn't stick to the bottom, then continue cooking until it is cooked to medium-rare, about 1 1⁄2 minutes more. Using a slotted spoon, transfer the foie gras to the bowl with the lamb, drain off all the fat, and return the skillet to the heat. Repeat with the remaining foie gras, draining off all but 1⁄4 cup fat from the skillet.
5. Add the duck to the skillet and cook, stirring occasionally, until warmed through, about 2 minutes. Using a slotted spoon, transfer the duck to the bowl with the lamb and foie gras, and return the skillet to the heat.
6. Add the chanterelles and cook, stirring, until they give off their liquid and are golden brown, about 4 minutes. Scrape the chanterelles and any fat left in the skillet into the bowl with the duck and foie gras, then stir in the Parmigiano and truffle juice, if using. Season the filling with salt and piment d'Espelette.
7. Heat the oven to 275°. In a large baking dish wide enough to fit all the tomatoes snugly, arrange tomatoes cut side up. Equally divide the filling among the tomatoes, then cover each with its reserved top. Drizzle the remaining 1⁄2 cup duck fat over and around the tomatoes and scatter the garlic, thyme, and bay leaf over the top. Bake until the tomatoes are tender and the filling is hot, 20 to 25 minutes.
8. Using a large spoon or spatula, gently transfer tomatoes to a serving platter. Pour the stock into the baking dish, scrape up any bits that are stuck to the pan, then pour the pan liquid, herbs, and garlic into a small saucepan. Stir the balsamic vinegar and lemon juice into the pan, bring to a boil, and cook until reduced to 1⁄3 cup, about 5 minutes. Pour the sauce through a fine sieve into a bowl, discarding the solids, and season with salt and piment d'Espelette to taste. Serve the tomatoes warm, with a spoonful of sauce on each, and extra sauce on the side.