Bring a saucepan of salted water to a boil, Add leeks, and cook until tender, about 4 minutes. Drain, and transfer to a blender. Add butter and season with salt and pepper. Puree; transfer to a warm place.
Place dried mushrooms in a spice mill or coffee grinder; grind to the consistency of coffee. Transfer to a plate.
Place egg in a bowl; add some salt and pepper, and beat lightly. Place flour on a plate. Dip lamb very lightly in flour; shake off excess. Dip into egg, then into mushrooms. Pat mushrooms to adhere so they coat the lamb heavily. Refrigerate up to 2 hours.
Heat oven to 500 degrees F.
Heat 2 tablespoons olive oil in a 10-inch skillet; add the shiitake mushrooms, garlic, and thyme. Cook, stirring occasionally, until mushrooms are tender, about 10 minutes.
Heat the remaining 2 tablespoons olive oil in an ovenproof skillet set over medium-high heat for 1 minute. Add the lamb; cook 2 minutes on one side, and then turn over. Place the skillet in the oven for 3 to 4 minutes for rare meat, or a little longer for more well done.
Let the lamb rest 1 minute, then cut it into 1/2-inch-thick slices. Place a dollop of leek puree on each plate, top with a portion of mushrooms, and then place the lamb on top. Sprinkle with a little coarse salt, and serve.
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