Special equipment: pastry brush
Preheat oven to 400 dgrees F.
In a large pot of salted water boil the potatoes until tender, about 5 minutes. Drain the potatoes and set aside in a large bowl.
In a large skillet melt 1 tablespoon of butter with 1 tablespoon of olive oil over medium heat and saute the onion and garlic until soft. Add the button mushrooms and saute until lightly browned. Add this mixture to the bowl with the potatoes.
Melt the remaining butter and oil in the same skillet and saute the shiitake mushrooms until golden, add to the potatoes. Pour the white vermouth into the hot skillet and simmer for 30 seconds. Add the vermouth, oregano, thyme, and parsley to the potato and mushroom mixture. Season, to taste, with kosher salt and pepper.
Crumble the soft chevre into the potato and mushroom mixture and gently combine.
Lay 1 sheet of phyllo on a clean, dry surface, and brush with clarified butter. Continue layering with 2 more sheets of phyllo brushing each sheet with clarified butter. Place the mushroom mixture across the lower third of the phyllo layers. Roll up the strudel like a spring roll, tucking in the sides to create a tight cylinder. Place the strudel on a parchment paper lined baking tray.
Brush the outside of the strudel with clarified butter and refrigerate for 30 minutes.
Bake for 15 minutes, reduce the heat to 350 degrees F. Bake for 15 minutes longer, until golden brown. Cut the strudel into 6 slices and serve with the warm Dill Sauce.
In a medium pot over medium high heat melt the butter and saute the shallots until golden. Add the cream and bring to a boil.
Reduce the heat to medium low and simmer until reduced by 1/3. Stir in the dill and season with the salt and pepper, to taste.