Remove the stems from the portobello mushrooms, and place the mushrooms face down in a baking dish. Drizzle lightly with oil, and season with salt and pepper. Roast at 190°C (Gas Mark 5 / 375°F) for around 20 minutes, until soft. Set aside to cool.
Meanwhile, add the pine nuts, parsley and garlic to a mini food processor, and blitz until finely chopped. Add the olive oil and a pinch of salt and pepper, and blitz again to finish the pesto.
Cut two squares from the puff pastry, each slightly larger than one of the cooked portobello mushrooms. Spread some of the parsley pesto in the middle of the square of pastry, and add the portobello mushroom. Top with some crumbled goat's cheese.
Bring the corners of the pastry squares up to meet in the centre, over the top of the portobello. Pinch the corners together so that the pastry doesn't flatten out again as it cooks.
Place on a lined, greased baking tray and return to the oven for a further 20-25 minutes, until the pastry is golden brown and crispy.