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.