Cut the stalks and fronds off the fennel bulbs, then cut the trimmed bulbs in slices or, if you prefer, into thinnish wedges. Add the fennel to a large sauté pan, large enough to hold them in one or two layers, and then add the water (or broth) and butter.

Cover the pan and simmer the fennel for 15-20 minutes (a bit more if cut into wedges) until the fennel is very tender and reduced in size, and the liquid has almost evaporated. If the fennel cooks before the liquid has evaporated, uncover and cook off the excess over a high flame. Let the fennel cool. Arrange the fennel neatly in a greased gratin dish, in alternating layers of fennel and grated cheese (and bechamel if using), ending with cheese. (Add any remaining liquid in the pan to the dish just before adding the final layer of béchamel and cheese.) Top with dabs of butter and a dusting of breadcrumbs, if you like.

Bake the dish in a hot oven (200°C/400°F) for about 15-20 minutes, until the top has formed a nice golden crust. (If you like a crustier top, you can run the dish under the broiler briefly until it has reached the stage you like, but be careful not to let it burn!)

Let your fennel gratin cool for a few minutes, then serve.