First make sure all your veggies are washed and cut into roughly the same size. In a large soup pot, pour in 6 cups of salted water and your potatoes. Bring to a boil, and cook until halfway done (check after 10 minutes and again at 15. Don't overcook.) Add the cauliflower in for the last minute or two. Drain and reserve the water (pour into a colander resting on another bowl to save it.)
Meanwhile, make the pistou. In a food processor, add in the mashed garlic, the tomato paste, the basil and a large pinch of salt. Pulse a few times, and then turn on to low and slowly drizzle in the olive oil. Process till smooth. Taste to adjust seasoning. Place in a small serving bowl with a spoon for passing.
In the now empty soup pot, add a little bit of olive oil or butter and turn on medium heat. Add back in the onions and the carrot, and stir to soften just 1 minute, and then the zucchini for an additional 2 minutes. Add in the potatoes, the cauliflower, the greens and the green beans and salt and pepper, and allow to soften up together just a few minutes, 3-5 more, adding a tsp of salt to the veggies and splash of vinegar to help deglaze any brown spots on the bottom of the pot.
Pour the potato cooking water back into the veggies and bring up to a boil. Reduce to a simmer, and taste the veggies to make sure they are all well cooked-- you still want to have a little bite to them. You are probably already done cooking at this point-- don't overcook or you'll have a bowl full of mush.
Ladle the soup out into individual bowls, and pass the pistou around. Everyone can stir in one large spoonful of the pistou into their own bowl (or more or less to taste.) Store any leftovers separately or mixed together in the fridge.