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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.