Place the carcass in a large soup pot or stockpot and add cold water to cover by 2 inches (about 14 cups). Bring to a boil over high heat, skimming any foam that rises to the surface.
Add carrots, celery, tomato paste, 1 teaspoon salt, and 1/2 teaspoon rosemary. Return to a boil; reduce to a simmer and cook until the broth is rich and flavorful, about 2 hours.
With a wire skimmer, remove and discard turkey carcass and vegetables. Strain and transfer 8 cups of broth to a large saucepan; freeze the remainder.
Bring broth to a boil over high heat. Add red-pepper flakes and remaining 1/2 teaspoon rosemary; season with salt. Add kale, reduce to a simmer, and cook until kale is tender, 5 to 7 minutes.
Stir in turkey, corn, beans, and scallions and cook just until corn is heated through, about 2 minutes. Season with salt if desired.