Melt the butter over low-heat, stir in the ginger root and remove from the heat. Allow the mixture to infuse for 2 to 3 hours or overnight for the flavor to develop.
In a small saucepan, heat the vinegar and whisk in the sugar until it is dissolved. Remove from the heat and all to cool. Stir in the salt, pepper, sesame seeds, and soy sauce, then whisk in the sesame and vegetable oils until the mixture is completely emulsified.
Reheat the butter and ginger root infusion until it is soft enough to spread easily. Brush each sheet of aluminum foil with some of this mixture, and sprinkle with a little salt and white pepper. With a very sharp knife, slice the tuna about 3/8 inch thick against the grain and lay 2 slices in a single layer on each piece of foil. Preheat a grill, griddle, or large heavy skillet to very high heat.
Toss the greens with just enough vinaigrette to coat all the leaves and reserve the remaining vinaigrette for another use. Mound an equal quantity of the salad on each of 6 serving plates.
Transfer the sheets of foil to the hot grill and cook for about 30 seconds, or until the tuna is seared on one side and still raw on the other. Immediately transfer the tuna slices, raw side up, to the plates and drape them loosely around the edges of the salad. Garnish with the chives and the shiitake caps, if desired, and serve immediately.
In a large heavy saucepan, heat the sesame oil over medium heat. Add the garlic, ginger root, and mushrooms and saute for 3 minutes, or until the mushrooms are just seared and still dry. Deglaze the pan with the vinegar and reduce over medium-high heat until almost no liquid remains. Stir in the sambal olek and salt and remove from the heat. Add the vinaigrette and stir to mix. Set aside and allow to marinate for at least 6 hours. The mushrooms can be kept covered in the refrigerator, for up to 4 days.