Croque Madame City Sliders

Ingredients:
Bechamel:
2 tablespoons unsalted butter
2 tablespoons flour
1 cup whole milk
Pinch of freshly grated nutmeg
1 bay leaf
1 thyme sprig
Kosher salt
¼ cup freshly grated Parmigiano Reggiano

Sliders:

8 city sliders, sliced in half lengthwise
16 slices ham, sliced thin
8 sliced Gruyere, sliced thin
2 tablespoons unsalted butter
8 quail eggs or small eggs
Kosher salt
Arugula and chopped chives, for serving

Directions:
Make the bechamel sauce:

In a small saucepan, melt butter over medium heat. Whisk in the flour and cook, stirring constantly, until you smell the flour and butter cooking, 2 to 3 minutes. Do not brown. Add the milk, nutmeg, bay leaf and thyme. Cook, stirring, until the mixture thickens, about 10 minutes. Remove the bay leaf and thyme. Stir in the Parmigiano. Transfer to a bowl to cool.

Prepare Sandwiches:

Using a 1 ½-inch wide ring mold, cut a hole in the center of each slider top. Melt butter in a large nonstick skillet. Place the top of each slider in the skillet, cut side down. Crack eggs into the center of the sliders, season with salt and cook until set. Meanwhile, spread a thin layer of bechamel on the cut side of the bottom slider buns. Top each bottom half with 2 slices ham and 1 slice cheese. Sandwich with top buns. Serve garnished with arugula and chives.