Heat sesame oil in a large pot over medium-high heat. Add onion and cook until softened, about 3 minutes. Add garlic and ginger and cook for another minute. Add vegetable broth, soy sauce, rice vinegar and chili garlic sauce (if using). Bring to a boil.
Add mushrooms and tofu to the pot. Reduce heat to medium-low and simmer for 10 minutes.
Divide soup between four bowls. Top with green onions and serve.