Preheat the oven to 180°C, gas mark 4. Put the grated potatoes and carrots in a bowl. Sprinkle with ½ tsp salt. Set aside for 10 minutes.
Squeeze out the excess water and transfer the vegetables to a separate bowl. Add the olive oil and spring onions and toss to coat. Season, then spread out on a baking tray.
Make 4 indents in the vegetables (for the eggs) and bake for 30 minutes, until golden and crisp. When 5 minutes’ cooking time remains, crack the eggs into the indents and cook until done to your liking.
Scatter the rösti with the spinach and feta to serve.