Grate the courgettes using the coarse side of a box grater. Place the grated courgette into a large bowl, season generously with salt, and set aside for 15 minutes. This crucial step draws out excess moisture, ensuring your rösti stay crisp rather than soggy.
In a separate clean bowl, whisk together the egg, a pinch of salt and pepper, the chopped mint, grated Parmesan, and flour until you have a smooth, lump-free batter.
After 15 minutes, you will notice the courgettes have released a significant amount of liquid. Drain the courgettes thoroughly—using your hands, a clean kitchen towel, or cheesecloth to wring out as much water as possible. This is the secret to preventing your rösti from falling apart during frying.
Add the squeezed, dry courgette to the egg-and-flour mixture. Stir well until everything is fully combined and evenly coated.
Heat a generous splash of oil in a non-stick frying pan over a medium heat. For best results, aim for an oil temperature of around 350°F (175°C)—if the oil is too hot, the rösti will brown on the outside before the centre has a chance to cook through. Using a large ice-cream scoop is a handy trick for creating evenly sized portions.
Carefully place spoonfuls of the mixture into the hot pan, flattening them slightly with the back of the spoon. Fry for 3–4 minutes per side, or until they are golden brown and cooked through. You may need to do this in batches to avoid overcrowding the pan.
Transfer the cooked rösti to a plate lined with kitchen paper to drain any excess oil before serving.