In a medium skillet over medium high heat, heat 2 tablespoons oil. Add garlic and mushrooms and sauté for 5 minutes. Add soy sauce and chili garlic sauce and cook for 5 minutes more or until softened and slightly reduced. Let cool.
Working with one wonton wrapper at a time, fill with about 2 teaspoons of cooled mushroom filling in middle of circle. Lightly brush edges with water and fold over to make a half moon shape, pressing tightly to seal. Set aside and repeat to fill all the wontons.
In a large pot, heat remaining 2 tablespoons oil. Add carrots and sauté 4 minutes. Add green onion and cabbage and sauté 2 minutes more or until softened. Add broth, water and salt and bring to a boil. Reduce to a simmer and cook for 8 to 10 minutes or until vegetables are tender. Add prepared wontons and cook 4 to 6 minutes or until wontons float and are tender. Divide between 4 bowls.