Any combination of wild mushrooms can be used; try chanterelles and shiitakes.
Split leeks in half lengthwise, leaving root attached. Remove some of the inner layers to use for tying purses; blanch them for 30 seconds in boiling water, refresh in cool water, drain, and set aside.
Remove green tops and thinly slice leeks. Saute in 1 tablespoon of the oil until soft, about 5 minutes. Do not brown. Remove from pan to a large bowl.
Add another tablespoon of oil to pan, turn heat to medium high, and add enough mushrooms to cover bottom of pan. Cook, stirring frequently, until just softened, about 3 minutes. Remove from pan and add to leeks. Repeat until all mushrooms are cooked. If using a variety of mushrooms, cook each type separately.
Return mushrooms and leeks to pan. Add thyme, salt, pepper, and Madeira or stock, and cook over high heat, stirring, until liquid is reduced to a syrupy consistency. Set aside to cool.
Unwrap phyllo pastry and unroll flat. Cover completely with damp dish towel to prevent drying out. Lay out a sheet of phyllo and brush it sparingly with melted butter. Repeat until you have three layers. Cut into 4 rectangles. Place 1/3 cup filling in the center of each and gather into a cinched-purse shape. Repeat with remaining filling. (Wrap and refreeze remaining phyllo dough.) Chill purses until firm.
Heat oven to 400 degrees. Place purses on a baking sheet and bake until evenly golden brown, about 15 minutes. Tie each purse with a leek ribbon and serve hot.