Preheat oven to 200ºC (390ºF). Brush the chicken with oil and sprinkle with salt and pepper. Heat a non-stick frying an over high heat. Cook the chicken for 2 minutes each side or until just brown. Remove the chicken from the pan and set aside. Add the mushroom and thyme to the pan and cook for 2 minutes or until tender. Add the parmesan, stir to combine and set aside.
Layer 5 sheets of the filo together, brushing between each layer with melted butter. Place a chicken breast on the combined filo sheets and top with some of the mushroom mixture. Roll filo to enclose the chicken and mushroom filling. Repeat with remaining ingredients to make 6 parcels.
Place on a baking tray lined with non-stick baking paper and bake for 20 minutes or until puffed and golden. Serves 6.