Preheat the oven to 375°F (190°C). Begin making the meat sauce: heat the oil in a saucepan over medium-high heat, add the ground beef, and cook, stirring frequently and breaking up any lumps, until the meat is evenly browned all over.
Sprinkle the flour over the browned meat and stir continuously for 1 minute to cook out the raw flour taste. Then add the stock, tomatoes, celery, onions, garlic, tomato paste, and sugar, stirring everything together well.
Season the sauce generously with salt and pepper, then bring it to a boil. Once boiling, reduce the heat, cover the pan, and let it simmer gently for 1 hour, stirring occasionally, until the sauce is rich and thickened.
While the meat sauce simmers, prepare the white sauce: melt the butter in a separate saucepan over medium heat. Sprinkle in the flour and cook, stirring constantly, for 1 minute to create a smooth roux.
Remove the pan from the heat and gradually blend in the milk, whisking continuously to prevent lumps from forming. Return to the heat and bring to a gentle boil, stirring constantly until the sauce thickens to a creamy consistency.
Let the white sauce simmer for 2–3 minutes to cook through fully. Stir in the mustard and a pinch of nutmeg, then season with salt and pepper to taste.
To assemble the lasagna, spoon one-third of the meat sauce into the bottom of a large shallow ovenproof dish, spreading it out evenly. Pour one-third of the white sauce over the meat sauce, then sprinkle with one-third of the combined Cheddar and Parmesan cheeses.
Arrange half of the lasagna sheets in a single layer over the cheese, breaking them if necessary to fit the dish. Repeat the layering process once more with another third of each sauce and cheese, then top with the remaining lasagna sheets.
Finish by topping with the final third of both sauces and the remaining Cheddar and Parmesan cheeses, spreading everything evenly to the edges.
Place the dish in the preheated oven and bake for 45–60 minutes until the pasta is completely tender when pierced with a knife and the topping has turned a beautiful golden brown color. Let it rest for 5 minutes, then serve hot, garnished with a sprinkle of fresh chopped parsley.