Preheat the oven to 220°C, fan 200°C, gas mark 7.
Place the potato wedges in a pan of cold salted water. Bring to the boil and simmer for 5 minutes, then drain.
Place the wedges on a large baking tray. Add 1 tablespoon of the oil and the chilli powder, season well with salt and freshly ground black pepper, and toss to coat.
Bake for 35-40 minutes, until golden, turning halfway through.
When the wedges are nearly done, heat the remaining 2 tablespoons of the oil in a non-stick pan, crack in the eggs and fry until the white is set. Serve the fried eggs with the sliced ham and wedges.