Slice the corn kernels off the cob and reserve the cobs. Heat 4 tablespoons butter in a large pot over high heat. Add the thyme, onion, celery, garlic, 1 teaspoon salt and all but 1/2 cup each corn kernels and ham; cook 1 minute. Add the potatoes, corn cobs and 5 cups water. Cover and cook until the potatoes are very tender, 20 to 25 minutes.
Meanwhile, heat the remaining 1 tablespoon butter in a skillet over medium-high heat. Add the reserved corn kernels and ham and cook until the ham is browned, about 6 minutes. Add the scallions and cook until softened, about 1 minute.
Remove the corn cobs from the soup and discard. Stir in the heavy cream and cook until heated through, about 1 minute. Ladle the soup into bowls and top with the corn-ham mixture.