Cook the potatoes in a saucepan of salted boiling water for 15 minutes or until tender. Drain well, return to the pan and leave to cool for 10 minutes.
Add 2 tablespoons of the oil, the egg and half the grated Parmesan to the potato and mix well. Turn out on to an oiled baking sheet and spread out to form a 25 cm (10 inch) round. Place in a preheated oven, 200°C (400°F), Gas Mark 6, for 15 minutes.
Remove from the oven and spread with the tomato paste or ketchup. Arrange the tomato and mozzarella slices on top. Scatter with the remaining grated Parmesan, thyme, if using, and a little salt. Drizzle with the remaining oil.
Return to the oven for a further 15 minutes until the potato is crisp around the edges and the cheese is melting. Cut into generous wedges, garnish with thyme sprigs, if liked, and serve.