Preheat the oven to 200°C / 400°F / Gas mark 6.
Line a baking tray with greaseproof paper and set aside.
Peel and deseed the squash by running a spoon down the centre, then cut into thin discs about 5mm thick.
Place on the baking tray, coat with the oil and sprinkle with the chopped sage.
Bake for 25 minutes until soft.
Meanwhile, pour the milk into a pan and add the onion and sage.
Bring to a simmer, then remove from the heat and set aside for 10 minutes.
Melt 25g butter in a large frying pan over a medium heat and add the mushrooms.
Season generously with salt and pepper then fry for 6-8 minutes until soft. Increase the heat, stir in the garlic and fry for a further 30 seconds.
Remove from the heat and set aside.
Strain the milk through a sieve to discard the onion and sage.
Heat 25g butter in a small pan, then add the flour, stirring continuously.
Slowly whisk in the milk.
Simmer for 3-4 minutes, whisking continuously until smooth.
Season and pour in any cooking liquids from the mushrooms and set aside.
Rinse the spinach and place the wet leaves in a large pan.
Cover and cook over a high heat for 2-3 minutes until wilted, then remove from the pan, draining off any liquid.
To assemble the lasagne, spread 2 tablespoons of the sauce in the base of a baking dish.
Place a sheet of lasagne on top. Top with 2 tablespoons of béchamel, then spread with half the mushrooms and sprinkle with a quarter of the cheese.
Repeat with a sheet of lasagne, 2 tablespoons béchamel, all the spinach, the remaining mushrooms and a quarter of the cheese.
Then repeat with another sheet of lasagne, the béchamel, half of the squash and a generous sprinkling of cheese.
Finish with a final sheet of lasagne, and top with the last of the squash, the remaining béchamel and the remaining cheese.
Dot over a few squash discs to decorate if you like at this point.
Loosely cover the top with foil and bake for 30-35 minutes until golden, removing the foil halfway through.
Serve immediately with a green salad.