In a saucepan, bring the mushrooms and 3 cups of water to a boil. Simmer over moderate heat until the broth is reduced to 1 cup, for about 1 hour. Strain and discard the mushrooms. . . or keep them. :)

Meanwhile, in a small saucepan, bring the garlic and olive oil to a boil. Simmer over moderately low heat, stirring, until the garlic is very tender and golden, about 30 minutes.

In a large enameled cast-iron casserole or Dutch oven, bring the tomatoes and the garlic and oil to a boil. Add the mushroom broth and, using an immersion blender or potato masher, puree the sauce until smooth.

Add the mushrooms and olives (if you are using) and bring the sauce back to a boil, then simmer over moderately low heat until thickened, about 1 hour. Season the sauce with salt and pepper.

Cook the spaghetti in a large pot of salted boiling water until al dente. Drain the spaghetti and return to the pot. Add 2 cups of the sauce and cook, tossing, for 1 minute. Transfer the spaghetti to bowls, top with the basil and serve with grated Parmigiano-Reggiano cheese.