Stuffed Mushrooms

Name: Stuffed Mushrooms
Ingredients: 1 pound 2-inch-wide stuffing mushrooms (about 14 to 16), cleaned; 1 tablespoon unsalted butter; 3 tablespoons extra-virgin olive oil, divided, plus more as needed; 2 tablespoons finely chopped shallot; 1/4 teaspoon kosher salt, plus more as needed; 2 tablespoons Madeira; 1 1/4 teaspoon chopped fresh thyme leaves; 1/2 cup fresh breadcrumbs; 1/4 cup finely grated Parmesan; 1/4 cup creme fraiche; 2 tablespoons finely chopped fresh flat-leaf parsley leaves; Freshly ground black pepper; 2 tablespoons water