Heat butter in a large pan over medium heat. Add shiitake mushrooms and sauté until softened, about 2-3 minutes.
Add minced garlic and finely chopped green onions. Sauté another minute or two, until the garlic and green onions are softened.
Slowly add white wine to pan. Stir ingredients and gently scrap any mushroom, garlic, or onion bits sticking to the pan.
Add crab meat and dill. Stir all ingredients until well blended and the crab is heated through, just 2-3 minutes.
Remove from heat and let cool 10 minutes.
Preheat oven to 375 degrees.
Prepare phyllo dough. Gently remove from package and lay out onto a large surface. Follow these directions as to how to fill and prepare the phyllo dough. I used about a tablespoon of the crab and mushroom filling per phyllo dough triangle, set approximately 2 inches from the corner of each phyllo dough sheet before folding into triangles.
Bake at 375 degrees for 20-25 minutes, until the edges of each triangle are golden brown. Be sure to watch the oven carefully the last 5 minutes.
Allow to cool for 10 minutes before serving.
Optional: Serve with lemon wedges and tartar sauce.