Preheat the oven to 200°C/ 400°F/ gas 6
Mix the squash, mushrooms and red onion in the oil and toss together with the sage, salt and pepper.
Spread the vegetables out in a large roasting pan, and roast them at the top of the oven for 30 minutes, turning them once, until soft.
Remove from the oven and set aside to cool
For the sauce, melt the butter in a small heavy-based saucepan. Remove from the heat and whisk in the flour.
Whisk in the milk a little at a time until the sauce is smooth.
Return the pan to the heat and cook, stirring constantly, until the sauce has thickened.
Reduce the heat to low and continue to cook, stirring occasionally for 5 minutes.
Add two-thirds of the grated cheese season well and cook for a further 2-3 minutes until thick and creamy.
Reduce the temperature to 180°C/ 350°F/ gas 4.
To construct the lasagne, spread one-third of the sauce at the bottom of a 20cm (8in) square ovenproof dish.
Cover with the half the vegetables, then a layer of lasagne.
Repeat the process, finishing with a layer of sauce on top.
Sprinkle the remaining cheese over the top and bake in the oven for 45 minutes – 1 hour, until it is well browned on top and soft when pierced with a knife.
Remove from the oven and allow it to rest for 5 minutes before serving.