In a medium bowl, mix together the mushrooms, cabbage, carrots, garlic, green onion, ginger, soy sauce, sesame oil, and white pepper to make the wonton filling.
Take one wonton wrapper and place 1 teaspoon of the filling in the center. Wet the edges of the wrapper with water, fold the wrapper in half to form a triangle, and press the edges together to seal. Repeat with the remaining wonton wrappers and filling.
In a large pot, bring the vegetable broth to a boil. Add the garlic, ginger, soy sauce, rice vinegar, and sesame oil and reduce the heat to low. Carefully add the wontons to the pot and simmer for about 5 minutes, or until the wontons are cooked through.
Season the soup with salt to taste.
Divide the soup and wontons among serving bowls, and garnish with chopped green onions.