Heat the olive oil in a large stock pot over medium-high heat. Add the shallots and about 3 pinches of salt and cook, stirring, until the vegetables soften, about 5 minutes. Add the thyme, water, 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. Strain the stock and discard the solids.
Yield: 4 cups stock
* Save the shiitake tops for later use.