Bring a medium saucepan of water to a boil. Add the snow peas and cook until crisp-tender, about 4 minutes. Fill a bowl with ice water. Drain the snow peas and transfer to the ice bath to cool; drain and set aside.
Whisk the cornstarch, soy sauce, sesame oil and 1/4 cup water in a small bowl; set aside.
Heat the peanut oil in a Dutch oven or heavy-bottomed pot over high heat. Add the scallion whites, ginger, carrots, sugar and 3/4 teaspoon salt; stir-fry 1 minute. Add the mushrooms and ham and stir-fry 1 more minute.
Add the rice wine, broth and 2 cups water to the pot and bring to a rapid simmer. Add the shrimp and cook, stirring, until pink, about 1 minute. Whisk the cornstarch mixture and stir it into the soup; simmer until slightly thickened, about 2 minutes. Stir in the snow peas and rice and season with white pepper. Divide among bowls; top with the scallion greens.