Preheat your oven to 200°C/Fan 180°C/Gas 6. Slice the top and bottom off the butternut squash. Peel the squash, then slice it in half and use a spoon to scoop out the seeds (see our tip on what to do with your leftover seeds). Chop the squash into small chunks and scatter them into a large roasting tin. Sprinkle with salt and pepper and drizzle over 1 tbsp oil, then toss to coat. Slide the squash into the oven and roast for 45-50 mins, turning once or twice, till tender and caramelised.
Scatter the cooked chestnuts into a second roasting tin. Thickly slice the mushrooms. Peel the red onions and slice them into thin wedges. Peel and thinly slice the garlic cloves. Pick the thyme leaves off their stems. Add the mushrooms, red onions, garlic and thyme to the chestnuts. Pour over 1 tbsp oil and add a pinch of salt and pepper. Toss the vegetables to coat in the oil, then slide onto the oven shelf below the butternut squash. Roast for 20-25 mins, till tender.
Meanwhile, make the pastry. Tip 300g of the plain flour into a large mixing bowl. Grate in 100g of the cheddar. Measure out 100g of the cold butter and cut it into small cubes. Add to the bowl. Use your fingertips to rub the flour, cheese and butter together till the mixture resembles breadcrumbs. Separate 1 egg yolk from the white (keep the white in a small bowl for later). Add the egg yolk and 6 tbsp water to the mixing bowl and stir together to make a dough. Add 1-2 tbsp more water if the pastry still looks a little dry. Wrap the pastry in greaseproof paper or beeswax food wrap and pop in the fridge to rest for 30 mins.
When the butternut squash is tender and caramelised, scoop half of it into a large bowl. Mash the half in the bowl with a fork. Strip the kale leaves from their thick stalks. Finely shred the kale leaves and add to the bowl with the squash. Grate the remaining cheddar into the bowl and scoop in half of the tub of crème fraîche. Add 1 tbsp wholegrain mustard and a little salt and pepper. Crack in the remaining whole egg and stir together well.
Fold the remaining whole chunks of squash, along with the vegetables from the other roasting tin into the bowl with the mashed squash and cheddar. This is your galette filling.
Lay the rested pastry onto a large sheet of greaseproof paper. Roll the pastry out into a large rectangle, approximately 25cm x 35cm, using the reserved 50g flour to dust your rolling pin if needed. Slide the rolled pastry and greaseproof paper onto a large baking tray.
Scoop the vegetable and cheese filling onto the centre of the pastry, and spread out, leaving a gap of approx 5cm around the edges. Fold the pastry up around the filling to create free standing sides. Brush the pastry sides all over with the reserved egg white.
Slide the galette into the oven and bake for 40-50 mins, till the pastry is dark golden in colour and very crisp. Allow to cool slightly before serving.