Rinse the leeks if necessary (see below) and trim off the root end and green tops. Cut the stalks into thin slices.

Sauté the leek slices in the butter, butter and oil mixture or the oil over gentle heat until they are well reduced and quite tender, without browning, about 15-20 minutes. Season the leeks with salt and pepper just after you add them to the skillet.

Sprinkle the flour and mix it in well, letting it sauté as well for a minute or two. Then add the broth, mix it well to fully incorporate the flour into the liquid, and let simmer, semi-covered, for another 15-20 minutes.

Meanwhile, toast your slices of bread and place each slice in the bottom of a soup bowl for each diner. Sprinkle the slices with some grated cheese. When the soup is done simmering, ladle over a nice serving into the soup bowls. Sprinkle over some more cheese and top with a nice grinding of black pepper and—if you like—a drizzle of best quality olive oil.

You can serve this soup immediately, but it is best after a few minutes' rest, which allows the soup to soak into the bread.