Peel and slice the shallots and 1 of the garlic cloves. Trim and finely dice the celery and fennel. Scrub the new potatoes and roughly chop them.
Heat 1 tbsp olive oil in a heavy based pan. Add the shallots, garlic, celery, fennel and potatoes. Cook on a low heat for 10 mins till the veg begins to soften. Stir often.
Pour 800 ml boiling water into a jug and crumble in the stock cube. Pour into the pan and simmer for 5 mins.
Cut the French beans into thirds, trimming the ends. Pod the broad beans and peas. Add them all to the pan. Cook for 2 mins without the lid on.
Pull the leaves from the mint sprigs and roughly chop them. Zest and juice the lemon. Peel and chop the remaining garlic clove. Pop into a food processor with 1 tbsp olive oil. Blitz to a smooth paste. Pop into a bowl.
Using a ladle, carefully decant 2 ladles worth of the soup into a food processor (you don’t need to wash it in between making your mint purée). Blitz till smooth. Pour back into the pan. Stir.
Season the soup and ladle into warm bowls. Swirl through the mint purée. Top with torn basil leaves.