1.Preheat the oven to 200C/Gas 6.
2. Roll out the puff pastry into a small rectangle about 15x20cm.Cut the rectangle into 4 equal pieces and lay them onto a baking tray lined with baking parchment.Cut a criss cross into the pastry for decoration.
3. Mix together the egg and milk and brush this over the puff pastry.Sprinkle over the grated cheese and a little pepper and bake in the oven for 20 minutes until risen and golden.
4.Meanwhile,set a large shallow sauté pan over a low heat and add the butter and oil.When the butter has melted,add the shallots,potatoes and carrots and fry them gently for 5 minutes until the shallots have softened.
5.Pour over the cider and bring the liquid to the boil.Leave to simmer for about 10 minutes,with a lid on until the potatoes are soft.
6. Add the rest of the vegetables,cream and tarragon and season with salt and pepper. Simmer for a couple of minutes until the vegetables are just cooked.
7. Serve straight away with the puff pastry top.