Place the porcini mushrooms in a fine sieve and submerge in warm water. Let sit for 5 minutes; they will plump up. Strain and set aside until ready to use.
Heat the olive oil in a large stock pot over medium-high heat. Add the onion, shallots, carrot, and celery, and stir to mix well. Add about 3 pinches of salt and cook, stirring, until the vegetables soften, about 5 minutes. Add the thyme, water, porcinis, and shiitake stems and bring to a high simmer over medium-high heat.
Immediately reduce the heat to a gentle simmer over medium-low heat and continue to cook until the liquid reduces slightly and the stock is flavorful, about 45 minutes. Remove the porcinis and reserve for cooking. Strain the stock and discard the solids.
Yield: 6 cups stock
* Save the shiitake tops for later use in the risotto.