Peel, seed, and cut enough ripe papaya into 1/2-inch cubes to make 1/2 cup; reserve for garnish. Cut remaining papayas into 1-inch chunks. In a blender or food processor, combine 1-inch papaya chunks, 1/2 teaspoon chili, and orange juice; purée until smooth.
Pour purée into a bowl; add broth, lime juice, shallots, and additional chili to taste; mix well.
Ladle into wide bowls. To each bowl, add equal portions shrimp, reserved ripe papaya cubes, diced green papaya, and avocado. Add salt to taste and sprinkle with cilantro.