For the Cheshire soup, bring the stock to the boil. Add the potatoes and simmer for 4-5 minutes, then add the carrots, leeks and oats and continue to boil for a further 4-5 minutes, or until the vegetables are tender.
Using a potato masher, carefully crush the vegetables in the pan until broken down.
Reduce the heat until the mixture is simmering. Add all of the cheese and stir until melted, then season, to taste, with salt and freshly ground black pepper. Continue to simmer for a further 3-4 minutes.
Meanwhile, for the fried cheese sarnies, spread one side of each slice of bread with a little of the butter. Sandwich each slice of cheese between two slices of bread, buttered-sides facing outwards.
Heat the remaining butter and all of the olive oil in a frying pan over a medium heat. When the butter is foaming, add one of the cheese sandwiches and fry for 2-3 minutes on both sides, or until the bread is toasted and golden-brown on both sides and the cheese is starting to melt. Remove from the pan and keep warm until ready to serve. Repeat the process with the remaining cheese sandwiches.
To serve, ladle the Cheshire soup into six to eight serving bowls. Sprinkle over the sliced leeks and watercress. Cut the fried cheese sarnies into squares and serve alongside.