1. Pour 1 tablespoon olive oil into a 12-inch nonstick frying pan over medium heat; when hot, add half of the shallots and garlic and stir until shallots are limp, 3 to 5 minutes. Add half of the mushrooms and stir until mushrooms begin to brown and any liquid is evaporated, 5 to 8 minutes. Add half of the vinegar and thyme; cook, stirring often, to blend flavors, 1 to 2 minutes longer. Pour into a bowl, repeat to cook remaining mushrooms, and pour into the bowl. Let cool to room temperature, about 25 minutes. Stir cheese and parsley into mushroom mixture; add salt and pepper to taste.
2. On a 12- by 24-inch piece of plastic wrap, lay one filo sheet flat (cover the remaining filo with plastic wrap to prevent drying) and brush lightly with melted butter. Top with another filo sheet and brush lightly with more butter. Repeat this process to stack all six sheets.
3. Spread mushroom mixture in a 3-inch band along one long side of dough, 2 inches in from the edge and the sides. Fold long edge and the ends of dough over filling. Gently lift plastic wrap under filled side of dough and guide it forward to form a compact roll, ending with seam down.
4. Gently transfer roll, seam down, to a buttered 14- by 17-inch baking sheet. Brush top with more melted butter; save any remaining for other uses.
5. Bake on the center rack in a 375° regular or convection oven until golden brown all over, 45 to 55 minutes. Serve warm, cut into 1 1/2-inch slices.