Potato Strudel with Dill Sauce

INGREDIENTS

• 2 tablespoons white vermouth
• 1 clove garlic, minced
• 1 tablespoon unsalted butter, plus 1 tablespoon
• 1 cup finely diced onion
• Kosher salt and freshly ground black pepper
• Dill Sauce, recipe follows
• 2 cups shiitake mushrooms, stemmed and sliced
• 2 tablespoons chopped fresh Italian parsley leaves
• 2 Yukon gold potatoes, cut into 1/4-inch cubes
• 2 tablespoons chopped fresh thyme leaves
• 2 tablespoons chopped fresh oregano leaves
• 3 sheets phyllo dough
• 1/2 cup soft chevre
• 4 cups button mushrooms, diced
• 1/2 cup clarified butter, melted
• Special equipment: pastry brush
• 1 tablespoon olive oil, plus 1 tablespoon