Day-After Turkey Soup

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Carcass of one turkey (about 1½ pounds) without skin

8 cups low-sodium chicken broth

1 head garlic, halved crosswise

2 bay leaves

2 sprigs fresh thyme

2 cups diced skinless turkey meat (white or dark meat)

½ pound green beans, trimmed and cut into 1-inch pieces

2 cups frozen corn kernels

1 large sweet potato, peeled and cut into ½-inch dice

1 cup small dry pasta shape, such as ditalini

¼ cup loosely packed fresh dill sprigs, chopped

½ teaspoon freshly ground black pepper

Kosher salt

Crusty bread for serving