The best mushrooms for this soup would be ceps, or white mushrooms - a Boletus species found in the woods in autumn. If you know what they look like they are easy to find and difficult to confuse with poisonous species. You can also occasionally pick them up at the greengrocers (if you can afford them). Otherwise use the ordinary 'shop' mushrooms.
I recommend frying the vegetables in olive oil.
1/2 kg mushrooms
1 medium cauliflower
1 large potato
1 onion
2-3 cloves of garlic
2 vegetable/mushroom stock cubes
A few black peppercorns, crushed
A sprig of fresh rosemary (or a pinch of dried)
Salt
Oil for frying
Remove the leaves and cut the cauliflower into large florets. Put it in a pot and add 4-5 cm. of water. Bring it to the boil and cover tightly. Turn the heat down a little and steam the cauliflower until it is soft enough to mash with a fork.
Chop the onion, potatoes and mushrooms into small pieces. Fry the onion first and add the mushrooms when the onions are transparent. Fry until the water has gone from the mushrooms then add the potatoes, crushed garlic and black peppercorns. Cover the pan and fry on a lowish heat for about 10 minutes.
Mash the cauliflower in its cooking water, until there are no big chunks left. Put everything into a 3 litres pot along with a litre of water, the rosemary and the stock cubes. Bring to the boil and simmer for about 20 minutes. Add more water if the soup seems too thick, taste it and throw a 1/4 teaspoon of salt in if you think it is necessary.