Preheat the oven to 400°F (200°C).

 Arrange the eggplant and potato slices on baking trays. Brush lightly with olive oil, season with salt and pepper, and roast for 20–25 minutes, turning halfway through. 

 Heat olive oil in a large skillet over medium heat. Sauté the onion and carrot for 5 minutes until softened. 

 Add the garlic and cook for 1 minute. 

 Stir in the tomato paste, crushed tomatoes, lentils, oregano, thyme, cinnamon, smoked paprika, salt, and pepper. Simmer for 15–20 minutes until thickened. Stir in the parsley. 

 For the béchamel, melt the vegan butter in a saucepan. Whisk in the flour and cook for 1 minute. 

 Gradually whisk in the plant-based milk until smooth. Cook until thickened, then stir in nutritional yeast, nutmeg, salt, and white pepper. 

 Grease a baking dish. Layer half of the potatoes, followed by half the eggplant, then spread the lentil filling evenly. Add the remaining eggplant and potatoes. 

 Pour the vegan béchamel sauce over the top and smooth with a spatula. 

 Bake for 35–40 minutes until the top is golden brown. 

 Let the moussaka rest for 15 minutes before slicing and serving.