Wash the leeks and slice them thinly. Then peel the potatoes and cut into cubes. Wash the broccoli, remove the stems and divide them into florets. Take a pan and soften the leeks in butter over a low heat for 10-15 minutes, then add the potatoes and broccoli, and sautè for a few minutes, stirring. Cover with the hot vegetable stock and continue cooking for about 30 minutes on low heat, add salt and pepper.
Set aside some florets of broccoli for garnish and then place the remaining mixture in a blender with the mint to obtain a 'vellutata'. Put the 'vellutata' back on the fire, season with salt and add a few leaves of mint. Serve the broccoli soup with a drizzle of extra virgin olive oil and you can also add croutons.