• Preheat Oven: Preheat your oven to 400ºF (200ºC) fan-forced.
• Grease the Baking Dish: Grease a large 9x13-inch (22cm x 33cm) rectangular baking dish with butter. Sprinkle some flour over the butter (edges and base) and set it aside.
• Melt Butter: Melt the butter (or oil if using) in a large deep skillet over medium-high heat.
• Sauté Onion: Add the chopped onion and sauté for 3 minutes until translucent.
• Add Garlic: Add the minced garlic and cook for 1 minute, until fragrant.
• Cook Vegetables: Add the chopped carrot, zucchini, and eggplant, sautéing on medium-high for 5 minutes while stirring occasionally.
• Add Tomatoes and Seasonings: Stir in the canned tomatoes, tomato puree, and seasonings. Bring the mixture to a boil, then reduce the heat to low-medium and simmer for 5 minutes, stirring occasionally.
• Simmer and Thicken: Leave the tomato mixture on low heat, stirring occasionally, to allow it to thicken while you prepare the béchamel sauce.

White Sauce

• Melt Butter: In a new medium-large skillet, melt the butter over low heat.
• Add Flour: Once melted, increase heat to medium and whisk in the flour. Cook for 1 minute while whisking.
• Add Milk Gradually: Pour the milk in small increments, whisking continuously to ensure the sauce stays smooth. Each time the mixture starts to thicken, add more milk and keep whisking. Continue this process until all the milk has been incorporated.
• Cook Until Thickened: Once all the milk is added, continue cooking over medium heat, whisking constantly until the sauce thickens, being careful not to let it boil.
• Season: Reduce heat to low, whisk in the nutmeg, and season with salt and pepper to taste.
• Add Cheese: Stir in the shredded cheese until melted and the sauce is smooth. Turn off the heat and set aside.

Assemble

• Layer Lasagna Sheets: Lay 4 lasagna sheets on the bottom of the greased baking dish.
• Add Vegetable Mixture: Spread half of the vegetable mixture evenly over the sheets.
• Pour White Sauce: Pour one-third of the white sauce over the vegetables and spread it evenly.
• Repeat Layers: Add another 4 lasagna sheets, spread the remaining half of the vegetable mixture, and another third of the white sauce.
• Final Layer: Place the final 4 lasagna sheets on top. Pour the remaining one-third of the white sauce evenly over the top, then sprinkle with the remaining shredded cheese.
• Bake: Bake for 35-40 minutes or until the top is golden and crispy. Remove from the oven and allow it to rest uncovered for 5-10 minutes before cutting and serving.
• Serve and Enjoy: Garnish with fresh parsley if desired. Enjoy!