Spread the potato cubes out in a roasting tin. Drizzle over half the oil, scatter over the thyme sprigs and season with salt and pepper. Bake in a preheated oven, 220°C (425°F), Gas Mark 7, for 10 minutes.
Stir the potato cubes well, then add the mushrooms and bake for 10 minutes. Add the tomatoes and bake for a further 10 minutes.
Make four hollows in between the vegetables and carefully break an egg into each hollow. Bake for 3–4 minutes until the eggs are set. Garnish the vegetable mixture with the parsley and serve straight from the tin, with buttered toast.