Heat grill to high. Brush mushroom caps on both sides with oil and season with salt and pepper. Grill, cap side down, until slightly charred, about 4 to 5 minutes. Turn the mushrooms over and continue grilling until just cooked through, about 3 to 4 minutes longer.
Stack the ingredients on each of 4 salad plates starting with a mushroom cap, a slice of cheese, some spinach, and a drizzle of Sherry Vinaigrette. Add another mushroom cap, more vinaigrette, spinach, and another slice of cheese. Place a third mushroom cap on top and finish by drizzling with additional vinaigrette and garnishing with chopped chives, if desired.