Preheat the oven to 200°C/400°F/gas mark 6. Line three baking trays with greaseproof paper. Divide the butternut squash slices between two baking trays in a single layer. Drizzle 1 tablespoon of the olive oil over each tray and season generously with salt and pepper. Cook in the oven for 25 minutes.
Lay the mushrooms on the remaining tray. Drizzle with 2 tablespoons oil. Sprinkle with thyme and garlic, and season with salt and pepper. Cook in the oven for 20 minutes.
Meanwhile, make a cheese sauce. Melt the butter in a saucepan over a medium heat. Add the flour and cook for 1 minute, stirring. Whisk in the milk until the sauce begins to thicken slightly. Stir in the nutmeg, nutritional yeast, Cheddar cheese and artichoke hearts. Add half the Parmesan cheese and stir until melted. Season with salt and pepper, and remove from heat.
Spread a thin layer of the sauce on the base of a deep 25 x 30cm baking dish to prevent the pasta sticking to the base. Lay a quarter of the lasagne sheets over the top and ladle a quarter of the sauce over. Spread out evenly and lay the butternut squash from one tray over the sauce.
Add another layer of pasta and a quarter of the sauce. Spread out evenly, then add the cooked mushrooms in a single layer. Add another layer of pasta, followed by a quarter of the sauce. Lay the remaining butternut squash over the sauce and spread the baby spinach leaves over the top.
Lastly, add the remaining lasagne sheets. Top with the remaining sauce and sprinkle with the remaining Parmesan. Cook in the centre of the oven for 40 minutes, then serve.