Place mushrooms in a plastic bag and fill with water. Seal bag and shake to wash mushrooms. Drain well. Repeat if mushrooms are still gritty. Trim off and discard bruised stem ends (or tough shiitake stems). Cut mushrooms into 1/4-inch-thick slices.
In a 10- to 12-inch frying pan over high heat, combine 1 tablespoon olive oil and the butter. When butter is melted, add shallots and mushrooms. Stir often until juices evaporate and mushrooms begin to brown, 6 to 7 minutes. Add wine and stir to release browned bits; boil until wine evaporates, about 1 minute. Add parsley, and salt and pepper to taste. Keep warm.
In a large bowl, mix remaining olive oil and vinegar. Add salad mix and stir to blend with dressing. Mound greens onto plates and top with equal portions of mushroom mixture.
With a vegetable peeler, shave thin curls of parmesan onto salads. Add salt and pepper to taste.