Preheat the oven to 450°F. In a large bowl, toss the squash and mushrooms with 2 tbsp of the olive oil, 1 tsp salt and plenty of pepper. Spread the mixture on a baking sheet, setting aside the bowl for later use. Roast the vegetables, stirring every 10 minutes , until tender, 25 to 30 minutes .
Meanwhile, cut open the chilies with kitchen shears and discard the seeds and stem. Place the chilies in a bowl and cover them with boiling water. Let sit until soft, 8 to 10 minutes . Drain and finely chop.
In a medium skillet, melt the butter and the remaining 1 tbsp of oil over medium heat. Add the onions and 1/2 tsp of salt and cook, stirring, until the onions are softened, 5 to 7 minutes . Stir in the chilies and garlic and cook, stirring for another 3 to 4 minutes . Remove from the heat.
Return the roasted vegetables to the bowl and stir in the onion mixture. Season with salt and peppers.
Place the spinach, with some water stilling clinging to its leaves, in a large pot and cook, covered, over high heat until wilted, 1 to 2 minutes . Drain and rinse under cold running water until cool, then squeeze dry and coarsely chop. Transfer the spinach to the bowl with the roasted vegetables and mushrooms.
For the Bechamel Pour the milk into a large saucepan and bring to a boil. Immediately take the pan off the heat. Stir in the onion, shallots, sae and bay leaf, cover and let sit for 30 minutes . Strain the milk and discard the solids.
In a medium skillet, melt the butter over medium heat. Add the flour and cook, stirring constantly, until the roux is fragrant and a shade darker, about 3 minutes . Slowly whisk in the milk until smooth. Bring to a simmer, then reduce the heat and cook gently, stirring occasionally, until thick, about 20 minutes .
Preheat the oven to 400°F. Spread a quarter of the bechamel on the bottom of a 9 x 13 inch lasagna pan or gratin dish. Make one layer of noodles, slightly overlapping. Spread half of the roasted vegetables over the noodles and sprinkle with a quarter of the cheese. Make a second layer of pasta, and add another layer of bechamel, then add the rest of the vegetables and another layer of cheese. Top with pasta and remaining bechamel and cheese.
Butter a piece of parchment paper and lay it buttered side down over the lasagna. Cover the pan loosely with foil and bake for 40 minutes . Remove the foil and bake until the lasagna is golden brown and bubling, 10 to 1 minutes . Let rest for 10 minutes before cutting and serving.
Adapated from The Flexitarian Table by Peter Berley.