Gently warm the olive oil in a wide saucepan and add the garlic and chilli. Cook for a minute or two without colouring before adding the courgettes. Season well with salt and pepper.
Cook the courgettes over a medium low heat for 30 minutes, adding 300ml water after 20 minutes when the courgettes should have released some of their own water and will very soft and mushy. Cook for a little longer if they are not all quite soft.
Check for seasoning and blitz fror 20-30 seconds with a stick blender or in a processor. You want a thick soup with a rough consistency, not a smooth soup.
Serve the soup in warmed bowls, sprinkled with the mint and Pecorino and drizzled with a little more extra vigin olive oil.