If you want to seriously one-up your grilled cheese skills, take a page out of the cookbooks of les Français and turn your attention to the croque madame. This broiled ham-and-cheese sandwich is the ideal hangover remedy, complete with a runny fried egg on top. And at Michele’s, chef Matt Baker’s team has got the technique down pat.

The sandwich begins, in truly decadent French fashion, not with bread, but rather thick slices of brioche.

“We joke that butter is Chef Matt’s favorite ingredient,” says chef de cuisine Rachel Bindel. “Using rich, buttery brioche compliments the ham and cheese and creates a tender sandwich.”

Speaking of which, you’ll need two ounces each of ham and cheese; at Michèle’s, the sandwich appropriately features Paris ham, a style of cooked ham perfected in the French capital that’s often brined with spices like juniper, coriander and clove. Cheese-wise, there’s none better than Gruyère for its rich, nutty flavor.

“Using a Parisian ham and Gruyère is an ode to our French culinary upbringings,” says Bindel.

But not all elements of this sandwich are exactly as you would find them in a Parisian café.

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After buttering the bread on both sides and piling the ham and cheese high, the team breaks from tradition by first griddling the sandwich in a nonstick skillet to achieve optimal browning. Only then is the sandwich topped, not with the traditional béchamel, but rather with a three-cheese combo of Parmesan, mozzarella and even more Gruyère.

“Gruyère and Parmesan cheese are in the mix for flavor, while the mozzarella helps acquire the highly sought-after cheese pull,” says Bindel. “Without the bechamel, the flavor of the ham and cheese comes through more and leaves the egg to create a sauce for the sandwich.”

Ah yes, that egg — the very thing that transforms the sandwich from a croque monsieur to a madame. Once the cheese-topped sandwiches are slipped under the broiler, you’ll fry eggs in butter. In the time it takes for the white to set, the sandwiches will be golden brown and ready to be topped with this pièce de résistance, the runny yolk just waiting to bathe the sandwich in all its richness.

At Michele’s, the croque madame is served with fries and two condiments: a malt vinegar aioli and, to truly infuriate the French, ketchup. The resulting sandwich is “rich, indulgent and comforting” — the perfect thing for Sunday brunch, particularly the morning after a night out.

Croque Madame

Servings: 2

Ingredients

8 (½-ounce) slices Paris-style ham
8 (½-ounce) slices Gruyère cheese
4 slices brioche bread
½ cup shredded Gruyère cheese
½ cup shredded mozzarella cheese
¼ cup grated Parmesan cheese
2 eggs
2 tablespoons butter
Salt and pepper, to taste

Directions

1. Build each sandwich by layering two pieces of Gruyère cheese, four pieces of ham and two pieces of Gruyère cheese between the brioche bread. Generously butter the sandwich on both sides as you would for a grilled cheese.
2. In a nonstick skillet over medium heat, brown both sides of the sandwiches. Remove onto a sheet tray.
3. In a bowl combine the shredded Gruyère, mozzarella and Parmesan. Top each sandwich with half of the cheese mixture.
4. Broil the sandwiches for about 4-5 minutes, or until the cheese is gooey and browning.
5. While the sandwiches are browning, heat a nonstick skillet. Melt 2 tablespoons of butter. Crack the eggs into the pan, and season with salt and pepper. Reduce the heat to medium-low, and cook until the whites are set but the yolks are still runny.
6. Transfer each sandwich to a plate and top each with a sunny side-up egg. Serve with ketchup, fries and malt vinegar aioli.