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At least an hour before you want to make the tart, make the pastry. Tip the flour, thyme, butter and ½ tsp salt into a food processor and pulse to the texture of breadcrumbs. Add the egg yolk and milk and pulse again until it all starts to come together. Tip out and press into a ball of pastry. Wrap, then chill for at least 1 hr. Can be prepared up to 2 days ahead.
On a lightly floured surface, roll the pastry out and line an 11 x 33cm rectangular, fluted tart tin. (The pastry may break up as you work it, so patch over any broken bits.) Keep any excess trimmings. Chill the tart case in the fridge or freezer for 20 mins.
While the tart case is chilling, tip all the ingredients for the parfait into a clean food processor with a generous pinch of salt and a good grinding of pepper. Blitz until the mix is as smooth as it’s going to get, then push through a sieve into a jug and set aside.
Heat oven to 200C/fan 180C/gas 6. Line the tart case with greaseproof paper and baking beans, then bake on a baking tray for 15 mins. Remove from the oven and remove the paper and beans. Check for any small holes or cracks in the pastry and patch up with the reserved trimmings. Place back in the oven for 15 mins, until golden. Remove from the oven and lower the oven to 120C/fan 100C/ gas ½. When the oven has cooled suitably, slide the tart case back into it, then fill as far as you can with the parfait mix. Bake the tart for 20 mins until just set. Leave to cool, trim any untidy edges, then remove from tin.
To serve, cut the tart into 6 slices. Place on plates, then place small blobs of caramelised onions and a scattering of hazelnuts around the tart. Neatly place a few salad leaves around and drizzle everything with oil and vinegar. Arrange 2 fig quarters on top of each slice and serve.
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