Steak Croque Madame 9 Steps
To prepare the béchamel sauce, in a small sauce pot, melt the unsalted butter on medium heat then add in the all-purpose flour and mix to combine with a wooden spoon until an even paste-like consistency is formed. Add in the warm whole milk a few tablespoons at a time, stirring until smooth after each addition, until all of the milk is incorporated. Season lightly with the freshly rasped nutmeg as well as kosher salt. Set aside.
Slice the cooked steak into approximately 1/8-inch thick slices.
Begin assembling the sandwiches: lay 2 slices of sourdough bread down, divide the grainy Dijon mustard evenly onto both of these slices and spread evenly. Next, layer the sliced steak, then top each beef layer with 1 slice of Gruyère cheese.
On the remaining two slices of sourdough bread, spread approximately 1/4 of the entire béchamel sauce on each, turn this over onto the beef and cheese layers.
Heat an oven proof skillet on medium to medium-high heat and add in about 1⁄2 tablespoon of the unsalted butter and swirl until the entire bottom of the pan is coated. Carefully lay in the assembled sandwiches and rub the sandwiches around in the butter that is melted until it has all been absorbed by the bread slice and cook until golden brown. While this first side is cooking, spread about 1/4 tablespoon of the softened butter on each of the slices of bread that is facing upwards.
Once the first side of the sandwich is browned, carefully flip over the remaining buttered side and cook until golden brown.
Once the bottom side is golden brown, remove from the heat and divide the remaining béchamel sauce on the top of each sandwich and spread to cover. Top the béchamel with an additional slice of Gruyère and broil on low in a preheated oven until cheese is melted and bubbly, approximately 5-7 minutes.
While the sandwiches are broiling, heat another pan and cook the eggs in the remaining butter until desired doneness is reached, season with salt and pepper.
To finish the sandwiches: remove from the oven and top each with a cooked egg. Serve immediately with a side of arugula as well as hot sauce of choice.