Preheat oven to 350 degrees. In a hot skillet coated with oil, saute garlic, ginger, onions and season. Add shiitakes and stir. Season then deglaze with juice. Check for seasoning. Mix shiitakes with spinach. Toss with olive oil, vinegar and season. In a sprayed lasagne dish, spread some spinach mixture then top with noodles. Top with spinach mixture then some yogurt then more noodles. Repeat finishing with spinach on top and drizzle some olive oil. Sprinkle with cheese and bake in oven for 1 hour. Should be screeching hot. Broil if necessary to brown. Let rest 8 minutes before cutting.
For the Vinegar Syrup: In a non-reactive saucepan, combine vinegars and simmer. Reduce by 60 to 70 percent until a syrup is achieved. Keep in mind that the syrup will thicken as it cools. Reserve at room temperature.
In a hot saucepan coated with olive oil, caramelize garlic and onions. Season and sprinkle on sugar. When brown, about 6 to 8 minutes, add stock and bring to a boil. Check for seasoning, should be salty like blanching water. Add basil and spinach and blanch for 2 minutes. Transfer to blender and lift with butter. Verify flavor.
PLATING Sauce plate with sauce and zig-zag syrup. Slice square of lasagne and place on top. Garnish with vinegar syrup.