Preheat the oven to 400 degrees F. Heat 3 tablespoons olive oil in a large pot or Dutch oven over medium-high heat. Add the leeks, season with salt and cook, stirring, until very soft, 4 to 5 minutes. Add the zucchini, bay leaves and a pinch of salt; cook, stirring, until coated, about 2 minutes. Add 4 cups water, 1/2 teaspoon salt and a few grinds of pepper. Bring to a boil, reduce the heat and simmer until the zucchini is very tender, about 10 minutes. Discard the bay leaves.
Working in batches if necessary, transfer the zucchini mixture to a blender and puree with the buttermilk, 1 cup dill and 1/2 cup parsley. Pour into a large metal bowl set in a larger bowl of ice water. Let sit, stirring occasionally, until cooled, about 10 minutes. Season with salt if necessary.
Meanwhile, toss the bread cubes with 1 tablespoon olive oil and a pinch of salt; spread on a baking sheet. Bake until toasted, 10 to 12 minutes.
Combine the chicken, remaining 2 tablespoons olive oil, the lemon zest and juice, mustard and 1 tablespoon each chopped dill and parsley in a bowl. Season with salt and pepper and toss.
Divide the soup among bowls and top with the chicken and croutons. Sprinkle with the chopped dill and parsley.