First make the pesto: chop the garlic and crush together with the pine nuts using a pestle and mortar. Mix in the remaining ingredients, season with salt and pepper. Mix it into a rough paste, then set aside.
Slice the leek, dice the potatoes and zucchini, cut the green beans into 2 cm pieces. Drain the canned beans.
Heat the oil and butter in a saucepan. Soften the leek for 5 minutes, add the potato cubes, pour in vegetable broth, then simmer for 10 minutes. Add the remaining vegetables and cook for another 10 minutes.
Mix 2 Tbsp pesto into the soup and serve, with the remaining pesto on the side.