These are absolutely scrumptious, and can be made with either phyllo dough or puff pastry. I prefer to make these with puff pastry, as it's easier to work with. Cook the mushrooms in sherry rather than white wine. Personal preference only!
Melt butter in a skillet over medium heat; cook and stir mushrooms and garlic in the melted butter for 1 minute. Pour in white wine; cook and stir mixture until all liquid has evaporated, about 5 minutes. Season with salt. Remove from heat.
Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.
Cut each puff pastry sheet into 4 equal rectangles, creating 8 total. Spoon mushroom mixture into the center of each rectangle; sprinkle with Parmesan cheese. Gather all 4 corners of a rectangle together around the filling; pinch corners and edges of puff pastry together forming a seal. Repeat with remaining rectangles. Arrange bundles on the prepared baking sheet. Lightly press bundles so the bottoms are flat.
Bake in the preheated oven until bundles are golden brown, about 15 minutes.
If using phyllo dough, unroll dough. Cut stack in half. Working quickly, place 1 sheet on a flat surface and brush lightly with melted butter. Place another sheet on top and brush with butter. Repeat until you have 4 or 5 sheets of phyllo. Do NOT brush top layer with butter.