Madames, preheat the oven to broil. Place the waffles on a rimmed baking sheet. Top 4 of the waffles with 3 slices of ham and 2 tablespoons of shredded cheese each. Set aside. 5. To fry the eggs, heat the ghee in a cast-iron skillet over medium heat.
When hot, crack the 4 eggs into the pan and fry on one side for about 2 minutes, until the whites are cooked and the yolks are still runny. Place the waffles under the broiler at this point (see Step 6). Season the eggs with salt and pepper and remove the skillet from the heat. 6. While the eggs are cooking, broil the waffles for 1 to 2 minutes, until the cheese is melted and the waffles are warm.
To assemble the Croques Madames, place 1 ham and cheese–topped waffle on a serving plate. Top with a plain waffle, then add a sunny-side-up egg and smother with Mornay sauce. Sprinkle with freshly ground pepper and garnish with thyme leaves, if desired.
Repeat with the remaining waffles, eggs, and Mornay sauce. 8. Store plain waffles in an airtight container in the refrigerator for up to 3 days or in the freezer for up to 1 month. Reheat the waffles in a preheated 375°F oven or toaster oven for 3 minutes or until warmed through.
Store the Mornay sauce in an airtight container in the refrigerator for up to 3 days; it will thicken overnight. Reheat the sauce in a saucepan over medium heat, stirring, until warm, about 2 minutes. Add a few tablespoons of broth if the sauce is too thick.
The waffles and Mornay sauce can be made up to 3 days ahead and stored in separate airtight containers in the refrigerator. The waffles can be frozen for up to a month.