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Heat the olive oil in a heavy-based pan or casserole dish and gently fry the onion, celery, minced parsley stems and a pinch of salt, stirring often, until soft – about eight minutes.
Add the tomatoes, stir for a minute or two, then add the courgettes, green beans and potatoes, and a pinch of salt, stirring to prevent sticking and so each chunk glistens with oil.
Add one and a half litres of water and a parmesan rind, if you have one, then raise the heat so the soup nearly boils.
Reduce to a simmer for 30 minutes, or until the vegetables are tender.
Meanwhile, make the pesto. Pulse or pound the garlic, nuts and a pinch of salt, until they form a rough paste.
Add the parsley and basil leaves, and pulse again until creamy.
Gradually stir in the cheese and oil.
About 10 minutes before the end of the cooking time, add the drained white beans, then leave the minestrone to sit for at least 10 minutes before serving, passing around the pesto so everyone can add the amount they like.