Fill a large pot about halfway full with salted water, and bring to a boil. This will be the pot you'll cook the wontons in.
To Make the Broth
Meanwhile, in a separate large stockpot over medium-high heat, heat the sesame oil and add the ginger and garlic. Sauté, stirring constantly, for about 1 minute, just until the garlic and ginger have softened and are fragrant.
Slowly add the chicken broth and soy sauce; bring to a boil. Reduce the heat and simmer for 15 minutes. While the broth is simmering, make the wontons
To Make the Wontons
In a large bowl, mix together all filling ingredients. Have a small bowl of water and wonton wrappers ready.
Place a tablespoon of filling in the middle of each wonton wrapper. Wet the sides of the wonton wrapper with water.
Fold filled wonton wrapper into a triangle, and then pinch together the longest sides to create an envelope-like wonton.Continue filling and folding wonton wrappers until all of the filling has been used. Keep filled wontons on parchment paper, covered with a damp paper towel to keep the wrappers from drying.
In the large stockpot with boiling water, cook wontons in small batches for 4-6 minutes per batch. When cooked, place into a bowl, and continue until you're done making the wontons.
When the broth is ready, transfer the wontons to individual bowls. Ladle the hot broth over the wontons. Garnish with sliced green onions and serve immediately.
Notes
If using store-bought wontons: If frozen, thaw wontons to room temperature. Make the broth according to directions above. Divide wontons between individual bowls and ladle hot broth over the wontons. Make ahead: You can make the wontons and refrigerate them uncooked up to 24 hours before you make the broth. Place assembled, uncooked wontons on a baking sheet and cover with plastic wrap until you are ready to use.Leftovers: Leftovers can be refrigerated in an airtight container for up to 4 days. The broth can be frozen in a freezer-safe container, without the wontons, for up to two months.