Pistou sounds fancy, but it's just a mixture of basil, garlic, and olive oil. In this recipe, the basil is joined by chives and radish greens for an extra punch of flavor.
In a large pot, melt the butter over medium heat. Add the shallots and cook, stirring occasionally, until softened, about 3 minutes. Add the quartered radishes and the potato; season with salt and pepper. Cook, stirring often, until the vegetables soften, about 5 minutes. Add the stock and milk and bring to a boil over high heat. Reduce heat to medium and simmer the soup until the vegetables are tender, about 15 minutes
Meanwhile, in a medium bowl, whisk the oil, basil, chives, lemon zest and juice, garlic, and the radish greens; season. Set the radish greens pistou aside.
Transfer the soup to a blender. Puree the soup until creamy; season. Refrigerate until chilled, at least 2 hours.
Divide the chilled soup among 4 bowls. Drizzle with the radish greens pistou. Garnish with the matchstick-size radishes.