Prepare the mandarin vinaigrette: In a saucepan, combine all of the ingredients except the olive oil, and reduce over low heat to 1/2 cup. Remove from the heat and let cool. Stir in the olive oil and reserve.
Prepare the ginger vinaigrette: In a bowl, mix together all of the ingredients and correct the seasonings, as necessary. Reserve
Preheat a grill. Brush the steaks with some of the mandarin marinade and season them with salt and pepper. Brush the whole shiitake caps with some of the marinade and season with salt and pepper. Grill the steaks to medium-rare and set aside in warm spot. Grill the mushrooms until they are soft and set aside.
Toss the salad greens with the reserved ginger vinaigrette, adding any collected meat juices to the dressing, and set aside.
In a large saute pan, heat the butter until it is foamy. Add the shallot and the sliced shiitakes and saute for several minutes, or until the shallot is translucent and the mushrooms have given up most of their liquid. Season with salt and pepper, to taste.
Mound the salad greens in the center of 4 dinner plates. Slice the steak thinly on the diagonal and fan the slices around 1 side of the greens with the grilled mushroom caps. Arrange the mushrooms decoratively on the opposite side. Serve immediately.