Samphire (Salicornia europaea) is a variety of seaweed that grows near to the coast and in salt plains — you can buy it at fishmongeries. It’s delicious raw or cooked: if you want to get a feeling for the taste before trying the soup, boil it and eat it when it’s nice and tender.
Peel and roughly slice the potato, then put the samphire in a pan with some water and boil it so that it’s just soft enough to eat (a little al dente will hurt no one at this stage — the cooking will continue later). Meanwhile, sauté the potato in the butter, but make sure the pan is big because you’re going to be cooking the rest of the soup in there later.
Once the potato looks pretty nicely cooked (such that you could take it out of the pan and just eat it) turn off the heat but leave the potato in. Then, once the samphire is done as described above, drain it in a colander and shove it all in the potato pan. Turn the heat back on and continue to sauté while you make up the stock. If you have fresh fish stock, use that; otherwise, use a vegetable stock cube (fish stock cubes usually ruin the flavour).
Put the stock in the pan and bring to the boil, then turn down and simmer gently for a few minutes. Finally, liquidize the lot.
Put a splash of cream in and stir. If you want to store the soup, let it cool and keep it in the fridge or freezer.
Serve with an extra artistic swirl of cream on each bowl (and perhaps a sprig of samphire if you have any left over). Delicious.