Place beans in a small saucepan and cover with a generous amount of water. Bring to a boil and cook, covered, until beans are tender, about 1 hour. Drain and reserve. (To test a bean for doneness, choose a small one. Smaller beans take longer to cook because of their dense centers. The bean should taste creamy, rather than powdery, inside.)
Make Pesto by pureeing garlic, olive oil, and a few basil leaves in a blender. Gradually add remaining leaves until all are pureed. Transfer to a small bowl, add grated cheese, and stir. Set aside.Cut each vegetable into even 1/2-inch dice and set aside. To avoid discoloration, reserve potatoes in a bowl of cold water.
Bring chicken stock to a boil in a large stockpot. Add white beans, potatoes, carrot, and onion. Return to a boil, reduce to a simmer, and cook, uncovered, 15 minutes. Add remaining vegetables and bring back to a boil. Reduce to a simmer and cook an additional 10 minutes, uncovered. Season to taste. Serve soup with a dollop of pesto in each bowl.