1. Start by making the pastry. Mix 150g flour with a pinch of salt. Cut the fridge cold butter into 1-2cm-thick cubes. Add them to the flour and rub in with your fingertips till it makes fine breadcrumbs (if you have a food processor, you can whizz them together for a few secs).
2. Add 4 tbsp very cold water and stir with a fork to bring it together into a soft, silky (not sticky) dough. Add more water if it's not quite coming together. It should be smooth, but not wet.
3. Turn the dough out onto a floured surface. Pat into a rectangle, then roll out to make a rectangle 1-2cm thick.
4. Grate half the frozen butter onto the bottom two-thirds of the dough rectangle. Fold the top third down to the centre, then fold the bottom third up and over the top. Roll out into a rectangle. Rotate the rectangle 90°. Repeat the process with the remaining frozen butter. Roll into a rectangle and fold into a letter-like piece of dough.
5. Wrap up in a clean tea towel or food wrap. Pop into the fridge to rest for at least 1 hr. You can make the pastry the day before and leave it rest overnight, just make sure it is well wrapped.
6. When you're ready to make your tarte tatin, heat the oven to 200°C/Fan 180°C/Gas 6. Fill and boil your kettle.
7. Peel and trim the carrots, then slice them in half lengthways. Peel and thinly slice the onions.
8. Gently melt a third of the butter in a 24cm ovenproof frying pan. Add the onions with a good pinch of salt and cook on a very low heat for 15 mins, stirring occasionally, till very soft. Add 2 tbsp balsamic vinegar and ½ tbsp sugar and bubble for a few mins. Spoon into a bowl.
9. Place the pan back on the heat. Add the remaining butter, sugar and the cumin and dried chilli flakes. Place the carrots in the pan. Pour in 200ml hot water from the kettle and cook for 15 mins till the carrots are tender, slightly caramelised and the liquid has evaporated. Turn them over every now and then.
10. Remove the pan from the heat. Make sure the carrots are arranged in a single layer on the bottom of the pan. Spoon over the balsamic onions. Crumble over half the feta and sprinkle over most of the thyme leaves. If your pan isn’t ovenproof, lay the carrots on the bottom of a 24cm tart tin or dish and top with the onions, feta and thyme.
11. Dust your work surface with a little flour. Remove the pastry from the fridge and roll it out into a large circle – about 1-2cm bigger than your pan, tin or dish. Lay the pastry over the carrots and onions and tuck the edges in around the veg to make a crust (this will be the bottom of your tarte)
12. Slide the pan into the oven and bake for 30 mins till the pastry is golden and firm.
13. While the tarte tatin bakes, roughly chop the walnut halves and toast them in a dry frying pan for 2-3 mins till they smell nutty.
14. Remove the tarte tatin from the oven and leave to rest for 5 mins. Place a plate on top of the pastry, making sure it is bigger than the pan, and carefully flip the tarte out. Crumble over the remaining feta. Scatter over the remaining thyme leaves and the walnuts. Serve straight away.