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First, make the pastry. Tip the flour into a bowl with 1/ 2 tsp salt. Add the butter and rub in using your fingertips until it resembles breadcrumbs. Stir through the cheese with a cutlery knife. Add the egg yolk, drizzle over 1 tbsp cold water, then use the knife to stir it in until clumps of dough start to form. Tip onto a work surface and bring the dough together with your hands into a smooth ball. Alternatively, you can make the pastry in a food processor. Shape into a disc, wrap in cling film and chill for at least 20 mins.
Using a mandolin or a sharp knife, slice the sweet potatoes, courgettes and aubergines lengthways as thinly as possible. Brush the aubergine slices with lemon juice as you go to prevent them from turning brown. Put the sweet potato in a bowl with 2 tsp water, cover with cling film and cook in a microwave on high for 2 mins, then remove and leave to cool. Do the same with the courgettes and aubergines, but cook for just 30 secs, then set aside.
Take the pastry out of the fridge and roll out to the thickness of a £1 coin. Line a 23cm fluted tart tin with the pastry. Trim the sides with a pair of kitchen scissors, leaving an overhang of about 1cm. Chill for another 10 mins. Heat oven to 200C/180C fan/gas 6 and put a baking tray on the middle shelf.
When the pastry is cold and firm, line with baking parchment (scrunch it up first to make it more pliable), then fill with baking beans. Bake for 15 mins, then remove the beans and parchment and bake for 5 mins more until the pastry is biscuity.
Meanwhile, make the filling. Put the mascarpone, eggs, pesto, breadcrumbs and Gruyère in a bowl, season and mix well. Remove the pastry case from the oven, trim the sides with a small sharp knife so they’re flush with the top of the tin, then spread the filling over the base.
Reduce oven to 180C/160C fan/ gas 4. Drain any liquid from the vegetables, pat dry on kitchen paper, then season them all well. Stack a slice of sweet potato, courgette and aubergine on top of each other then, starting from one end, roll into a spiral. Put in the middle of the tart. Layer up another three vegetable slices, then wrap these around the spiralled veg in the centre. Continue until the tart is full and you have created a rose effect. Sprinkle thyme leaves between the layers and drizzle the tart with oil.
Bake in the centre of the oven for 40-45 mins until the vegetables are tender and the filling has set. Remove from the oven and leave to cool in the tin for 15 mins before removing. Serve warm or cold.