Heat large pot over medium heat. When pot is warm, add broth, green onions, soy sauce, salt, ground ginger, and pepper. Stir to incorporate ingredients, then taste broth and adjust seasonings as needed.
Bring mixture to boil, then immediately reduce heat to medium-low and let mixture simmer.
While broth simmers, crack eggs directly into small bowl. Whisk vigorously until eggs are fully scrambled. Egg mixture should be pale in color and no longer streaky.
When eggs are ready, use wooden spoon to stir broth in large circular motion. Stir multiple times in one direction to make liquid swirl gently around pot.
While liquid swirls, very slowly pour thin stream of whisked eggs into pot. Eggs will form thin ribbons upon contact. Continue pouring until all of egg mixture is added.
Once eggs have been added to broth, use spatula or slotted spoon to carefully add wontons to soup. When all wontons have been added, let soup simmer 5 to 8 minutes, or until wontons float to surface of soup.
When wontons are cooked through, portion soup into serving bowls. Garnish with additional chopped green onions if desired and serve warm.