Top and tail the courgettes, quarter them lengthways, then cut into 3mm-thick half-moons. Warm the oil in a large, heavy-based pan and fry the shallots until they are starting to soften. Add the courgettes and potato, and cook, stirring, for about five minutes.
Add 1.4 litres of water or light vegetable stock, bring to a boil, then reduce to a simmer and cook for 40 minutes, until the courgettes are extremely soft and collapsing.
Meanwhile, put the cubed bread on a baking tray, zigzag with olive oil and bake until lightly toasted and golden.
In a small bowl, whisk the eggs, parmesan, basil and a good amount of black pepper.
Pull the soup off the heat, wait a minute, then stir in the whisked egg mixture – it will curdle slightly and turn the soup cloudy. Divide between bowls, top with some croutons and serve.
