How to Make It
In a medium saucepan of boiling water, blanch the parsley leaves until tender but still bright green, about 3 minutes. Drain well. Transfer the parsley to a food processor and puree. Scrape the puree into a bowl. Clean out the food processor.
Add the white mushrooms to the food processor and pulse to mince. In a large nonstick skillet, heat the oil. Add the minced mushrooms, season with salt and pepper and cook over high heat until the liquid has evaporated, about 5 minutes. Add the water and stir to scrape up any browned bits. Add the cream, cover and simmer over low heat for 10 minutes. Strain the mushroom cream through a fine strainer into a small saucepan, pressing on the solids. Simmer the cream over moderately high heat until reduced to 1/3 cup, about 2 minutes. Season with salt.
In a large skillet, melt the butter. Add the shallot and cook over moderately high heat until softened, 2 minutes. Add the chanterelles, season with salt and pepper and cook over moderate heat, stirring occasionally, until browned, 10 minutes.
Set a round rack in a large, wide pot. Add enough water to reach just under the rack without touching it. Cover the pot and bring the water to a boil. Butter four 1 1/4-cup heatproof porcelain bowls, about 4 1/2 inches wide at the top tapering down to 2 inches wide at the bottom. Alternatively, use ramekins. Spoon the parsley puree into the bowls. Crack 1 egg into one of the porcelain bowls; season with salt and pepper. Repeat with the remaining eggs. Cover each bowl with plastic wrap. Carefully set on the rack in the pot, cover and steam over low heat until the whites are firm and the yolks are runny, about 12 minutes. Discard the plastic wrap.
Reheat the mushroom cream and chanterelles. Spoon the cream over the eggs, top with the chanterelles and serve.