In a large pot, sauté the shallots gently in olive oil and a knob of butter, just until soft and fragrant. Add the lentils and mix. Let the lentils simmer gently, mixing from time to time, for a minute or two.
Add the broth or water. Simmer for about an hour, or until perfectly tender, adding more liquid if needed. (Lentils, like all legumes, absorb a great deal of liquid as they cook.)
Purée the lentils until smooth. (For this, the easiest method is to use a hand mixer directly in the pot, but you can also transfer the lentils to a blender or food processor, or—in the old fashioned way— pass them through a food mill and return them to the pot.)
Mix in the heavy cream and simmer the purée for a few minutes longer, to allow the flavors to meld. Let it simmer if it needs to thicken up. If it's too thick for your taste, on the other hand, add some more broth or water.
While the lentils are cooking, brown the sliced mushrooms in garlic and oil, following our directions for making funghi trifolati. When they are nicely browned, remove them from the heat and sprinkle with the parsley.
In a separate small saucepan, very gently heat some olive oil (say around a quarter-cup) with a clove of garlic and a sprig of fresh rosemary. Let the oil infuse for about 2-3 minutes and turn off the heat.
To serve, lay most of the slices of mushrooms down at the bottom of soup tureen (or in individual bowls as pictured), then ladle over the lentil purée. Garnish the top with the rest of the mushrooms, then drizzle everything with the infused oil and sprinkle with fresh parsley for color if you like.
Serve with slices of grilled or fried bread.