How to Make It
Make the filling: Preheat oven to 400°F. In a large cast-iron skillet, melt 2 tablespoons butter over moderately high heat. Add mushrooms, thyme, and garlic. Transfer to preheated oven, and roast until mushrooms are tender and lightly browned, about 15 minutes. Stir in parsley and season with salt and pepper.
Make the sauce: In a medium saucepan, melt 1 tablespoon butter over moderate heat. Whisk in all-purpose flour until combined. Gradually whisk in 1/2 cup cream and simmer until thickened, about 2 minutes. Gradually whisk in remaining 1/2 cup cream, nutmeg, and 1/2 cup of the mushroom filling. Simmer over low heat, stirring occasionally, until no floury taste remains, about 7 minutes. Transfer to a food processor or blender and puree until smooth. Return sauce to pan and season with salt. Keep warm.
Make the crêpes: In a medium bowl, whisk 4 eggs, milk, bread flour, melted butter, and kosher salt until smooth. Heat a 10-inch nonstick skillet or crêpe pan over moderate heat; brush with melted butter. Add one-fourth of the batter, swirling to coat the pan evenly. Cook until lightly browned on bottom, about 2 minutes. Using a spatula, flip crêpe; reduce heat to moderately low. Arrange » one-fourth of the remaining mushroom filling in a ring on crêpe, and immediately crack 1 egg in center of ring, puncturing egg white so it spreads around crêpe. Cook until egg white is set and yolk is still runny, about 3 minutes. Transfer to a plate and top with a few mesclun leaves. Drizzle with some mushroom sauce, and garnish with thyme, salt, and pepper. Repeat procedure to make 3 more crepes. Serve immediately.