Croque Madame Burger

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 3/4 inches thick. Chill the patties for 1 hour in the fridge.
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.
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.
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.
To assemble, spread spicy brown mustard on the buns to taste. Layer the burger, bechamel sauce, and fried egg.