In a large cast-iron skillet, melt the butter over moderately high heat. Add mushrooms, thyme, and garlic and saute about 15 minutes, until caramelized. Stir in parsley and season with salt and pepper.

In a medium saucepan, melt the butter over medium heat. Whisk in the flour until combined. Gradually whisk in the cream and simmer until thickened, about 2-3 minutes. Add in the nutmeg and mushroom filling. Simmer over low heat, stirring occasionally, until no floury taste remains, about 7 minutes. Keep warm.

In a medium bowl, whisk 4 eggs, milk, flour, melted butter, and kosher salt until smooth. Heat a 10-inch nonstick skillet over medium 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 cheese and the mushroom mixture on crêpe, and cook for another minute or two. Transfer to a plate and top with a fried egg and a few baby market greens leaves. Garnish with thyme, salt, and pepper. Repeat as needed.