Add about 1-inch water to a medium saucepot over medium heat and bring to a simmer. Add the polenta to the chicken stock and stir to combine.
Place into a bowl and place over the simmering water to act as a double boiler. Once it starts to get soft, after about 5 minutes, add the mascarpone, 2 large tablespoons Parmesan cheese and 2 large tablespoons butter. Stir to combine and allow to slowly cook for about 40 to 50 minutes, stirring occasionally. If the polenta starts to become too hard, add more liquid, chicken stock or water, as needed.
Place a saute pan over medium heat and add 1 large tablespoon butter. Add the chanterelles and evenly space them in the pan to not overcrowd. Season the mushrooms with salt, to taste. Add the shallots, garlic and 6 to 8 sprigs thyme and allow the mushrooms to caramelize.
Place another medium saute pan over medium heat and add 1 heaping tablespoon unsalted butter and melt, not letting the butter foam or burn. Gently add the egg to the melted butter, season with salt and slowly cook, about 2 minutes. Repeat with second egg.
Place a spoonful of polenta in the middle of each dish. Top with an egg, surround with caramelized mushrooms, parsley leaves, shaved Parmesan, and shaved truffle, to taste. Season with sea salt, freshly ground black pepper and a drizzle of extra-virgin olive oil.
Rinse the chicken parts thoroughly. In a 10-quart stockpot, combine the onion, carrot, garlic, thyme, salt, bay leaf, peppercorns, and 1 gallon cold water. Bring the liquid to a boil over high heat, skimming any foam and impurities that rise to the surface. Reduce the heat to low and simmer for 5 hours, skimming the surface as necessary. Strain through a fine-mesh strainer, discarding the solids.
The double boiler helps to prevent the polenta from burning and overcooking.