How to Make It
Preheat the oven to 375°. In a very large skillet, melt the stick of butter. Add the shallots and cook over moderate heat until softened, about 4 minutes. Add the shiitake and white mushrooms, season with salt and pepper and cook over moderately high heat, stirring occasionally, until the liquid evaporates and the mushrooms are lightly browned, about 8 minutes. Add the sherry and sherry vinegar and cook for 1 minute. Season with salt and pepper.
Transfer the mushrooms to a food processor and process until finely chopped but still a bit chunky.
Lay 1 sheet of phyllo on a work surface and brush with melted butter, then top and repeat with 2 more phyllo sheets; keep the rest covered with plastic wrap. Halve the layered phyllo to form two 8 1/2-by-12-inch rectangles. Spoon 1/2 cup of the mushrooms at the bottom of each half, leaving a 1-inch border. Roll the phyllo over the mushrooms; fold in the sides and continue to roll up like a cigar. Brush each strudel with melted butter and transfer to a large baking sheet. Repeat with the remaining phyllo, mushrooms and butter.
Bake the strudels for 15 minutes, or until nicely browned and crisp. Let rest on the baking sheet for 3 minutes. With a serrated knife, cut the strudels on the bias into 2-inch lengths. Transfer to a platter, cut sides up, and top with small dollops of sour cream. Serve hot.
The mushroom filling can be refrigerated for up to 2 days. Bring to room temperature before filling the phyllo.
Clean mushrooms right before using them. Wipe them gently with a damp cloth or paper towel, or rinse them quickly under cold water and dry them immediately. Do not soak them in water, because they absorb liquid like sponges. For hard-to-reach or stubborn dirt, clean the mushrooms with a soft brush (a small paint brush works well).