Coarsely chop half of shiitake mushrooms and white mushrooms; then quarter remaining shiitake mushrooms and white mushrooms. Melt 2 tablespoons butter in Dutch oven over medium-high heat. Add all shiitake mushrooms and white mushrooms (both chopped and quartered) and 3/4 teaspoon salt. Cover and cook until mushrooms release their liquid, about 5 minutes. Uncover and continue to cook, stirring occasionally, until all liquid has evaporated and mushrooms begin to brown, about 10 minutes.
Add shallots, sage, garlic, and porcini mushrooms and cook until fragrant, about 1 minute. Add wine and cook until evaporated, about 2 minutes. Stir in 4 cups water, pasta, and 1 1/4 teaspoons salt and bring to boil. Reduce heat to medium, cover, and cook, stirring occasionally, until pasta is tender, 12 to 15 minutes.
Off heat, stir in Pecorino, 1/4 cup hot water, lemon juice, remaining 2 tablespoons butter, and 1/2 teaspoon pepper. Stir vigorously for 1 minute, until sauce is thickened. Season with salt and pepper to taste. Transfer to serving dish and sprinkle with chives. Serve, passing extra Pecorino separately.