Step 1Cut the shiitake caps into quarters. Cut the morels lengthwise into quarters. If the chanterelles are large, cut them to comparable size.
Step 2Melt the butter in a large, deep skillet over medium heat. Add the leeks and cook 2 minutes. And all the mushrooms and season with salt. Cook, stirring constantly, until the mushrooms start to soften. Continue cooking, stirring often, until the mushrooms give up their liquid, 10 minutes. Add the vermouth and continue cooking and stirring occasionally until almost all liquid has cooked away.
Step 3Add the stock and continue cooking, stirring occasionally, until it is mostly reduced, about 10 minutes. Add the cream and cook until the mushrooms are very tender and the liquid has thickened into a sauce. Add the herbs to taste and more salt if needed, then season with white pepper. Serve hot.