Croque Madame / Monsieur Burger

Directions

1. In a bowl, use your fingers to gently combine the ground beef, pork shoulder, salt and pepper. Make 4 patties by lightly forming 4 equal-size balls of meat and softly patting them until ¾ inches thick. Chill the patties for 1 hour in the fridge.
2. For the peppered bechamel sauce, heat a saucepan over medium heat and and the butter. When melted, add the flour and stir with a whisk until smooth. Cook while constantly stirring until the roux (combination of flour and fat) becomes a golden blonde color. Meanwhile, in a separate pan, heat the milk until it is almost boiling. Whisk the milk into the roux 1 cup at a time until very smooth. Bring the mixture to a boil and cook for 10 minutes while constantly stirring. Remove from heat and gently stir in the salt, pepper, and nutmeg. Set aside.
3. Heat a cast-iron skillet over medium-high heat and coat it with 1 tablespoon of the oil. Sear and cook the patties until a crust forms, or about 4 minutes per side. Top each patty with a slice of cheese during the last minute.
4. In another pan, fry the eggs in 1 tablespoon of olive oil over a medium-low heat to desired doneness. Personally, I’m a runny yolk kinda gal, but to each their own. In a third pan over medium-high heat, toast the inside of the buns until lightly brown.
5. To assemble, spread spicy brown mustard on the buns to taste. Layer the burger, bechamel sauce, and fried egg.