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.