Chef John's Creamy Mushroom Pasta

Ingredients

• ounces fettuccine pasta
• tablespoons olive oil
• pound fresh white mushrooms, sliced
• pound fresh shiitake mushrooms, stemmed and sliced
• salt and ground black pepper to taste
• garlic, minced
• fluid ounces sherry
• chicken stock
• heavy whipping cream
• teaspoons chopped fresh thyme
• teaspoons chopped fresh chives
• teaspoons chopped fresh tarragon
• tablespoons freshly shredded Parmigiano-Reggiano cheese, divided