Mash the filets of anchovies in the bottom of a shallow baking dish (save oil for a salad dressing). Mix the cream with them, and put the dish in a hot oven.
Stir two or three times after it has started to bubble, turning in the golden crust. Add the mushrooms and the parsley.
When reduced about one-third, turn off the oven. Remove the dish, and break the eggs carefully into it. Put the cheese over them, and the pepper. Then put back into the lowest part of the oven, and when the gentle heat has made the eggs firm but not hard, usually in about 15 minutes, remove and serve.
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