Preheat the oven to 350°F (175°C). Lightly grease a 9×13-inch casserole dish.
In a large skillet over medium heat, cook the ground beef until browned and cooked through, breaking it up with a spoon. Drain excess fat and transfer to a large mixing bowl.
In the same skillet, add the cubed chicken breasts. Season with salt, black pepper, garlic powder, and onion powder. Cook over medium heat until the chicken is cooked through and no longer pink inside, about 6-8 minutes. Add the cooked chicken to the bowl with the ground beef.
Add the diced ham, shredded Swiss cheese, and shredded mozzarella cheese to the meat mixture. Stir to combine evenly.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until lightly golden, stirring constantly.
Gradually whisk in the milk, ensuring no lumps form. Continue to cook and whisk until the sauce thickens, about 5 minutes.
Remove the sauce from heat and stir in the Dijon mustard, dried thyme, and nutmeg. Season with salt and pepper to taste.
Pour the sauce over the meat and cheese mixture and stir gently until everything is well coated.
Transfer the mixture into the prepared casserole dish and spread evenly.
Sprinkle the crushed buttery crackers evenly over the top of the casserole.
Bake in the preheated oven for 30-40 minutes, or until the casserole is bubbly and the topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.