Combine rinsed rice, broth, ginger, and salt in a large pot; boil over high, reduce to low, and simmer partially covered.
Cook 40-45 minutes, stirring occasionally to prevent sticking, until rice breaks down into creamy porridge. Add water if too thick.
Meanwhile, soft-boil eggs 6-7 minutes; transfer to an ice bath, peel, and set aside.
Remove ginger; stir in sesame oil and adjust salt to taste.
Ladle into bowls; top each with a halved egg, green onions, crispy shallots, and soy sauce. Let everyone add favorite toppings.