How to Make It
In a food processor, puree 3 of the bell peppers until smooth. Strain the puree through a fine sieve into a small bowl, pressing on the solids; you should have about 1 cup of pepper juice.
In a large saucepan, combine the remaining 3 bell peppers with the piquillo peppers and vegetable stock. Bring to a boil and simmer over moderately high heat until the peppers are tender, about 15 minutes. Transfer to a food processor and puree until smooth. Strain the puree through a fine sieve into a large bowl, pressing on the solids. Add the fresh pepper juice to the bowl and stir to blend. Season the soup with salt and lime juice, cover partially and refrigerate until cold, about 3 hours.
Meanwhile, prepare a large bowl of ice water and bring a medium saucepan of water to a boil. Using a sharp paring knife, score an “X” on the bottom of the tomato and blanch in the boiling water for 30 seconds. Using a slotted spoon, transfer the tomato to the ice bath. Add the green beans to the boiling water and cook until just tender, 3 to 4 minutes; using the slotted spoon, transfer them to the ice bath. Add the cranberry beans to the boiling water and cook until tender, 15 to 20 minutes. Drain the cranberry beans and transfer them to the ice bath.
Peel, seed and chop the tomato; transfer to a medium bowl. Cut the green beans into 1-inch pieces and add to the bowl, then add the cranberry beans. Stir in the lemon juice, olive oil, mint and preserved lemon and season the bean salad with salt and freshly ground pepper. Toss to coat.
Spoon the bean salad into the center of shallow soup bowls, pour in the chilled piquillo soup and serve.