Preheat the oven to 180C/350F/Gas 4.
For the tarte tatin, place the flour, chopped dill, salt, freshly ground black pepper and butter into a food processor and pulse until the mixture resembles breadcrumbs. Add the water a little at a time until the mix starts coming together to form a smooth dough. Knead into a ball and then roll out on a floured work suface to the thickness of a pound coin.
For the balsamic glaze, melt the butter in an ovenproof frying pan over a medium heat, add the balsamic vinegar and chilli flakes and bring to a gentle simmer. Add the tomatoes and stir in the balsamic sauce until coated.
Gently place the pastry over the top of the pan, pressing down the edges (inside the pan) to seal around the tomatoes, then cut off the excess pastry. Brush the pastry top with the beaten egg and then bake for 12-15 minutes, or until the pastry is golden-brown and well risen.
To serve, leave the pan to rest for a few minutes once it is out of the oven. Slide a sharp knife around the inside of the pan, place a plate on top and carefully invert the pan to turn out the tarte tatin. Serve in slices while still warm.