Set frozen puff pastry sheets on a counter and thaw for 30 minutes. Preheat oven to 375 degrees F (190 degrees C). Grease two 12-cup muffin tins. Melt butter in a large skillet over medium-high heat. Add white mushrooms and portobello mushrooms. Cook, stirring occasionally and adding garlic halfway through, until liquid evaporates, about 5 minutes. Add Italian seasoning and pepper. Unfold pastry sheets onto a lightly floured surface; roll each into a 12x16-inch rectangle. Cut each rectangle into twelve 4-inch squares. Drape squares into muffin cups. Place 1 1/2 teaspoon mozzarella cheese into each cup and spoon 1 1/2 teaspoon mushroom mixture on top. Add a portion of Parmesan cheese. Gather the 4 edges of the square to the middle; twist to seal the puff and seal the edges. Repeat until cups are filled. Bake in the preheated oven until golden brown, about 20 minutes.