To make the wonton strips, heat the oil in a heavy skillet over medium heat. Add the wonton strips to the skillet and cook until golden brown, turning over to brown both sides evenly. Remove from the oil and drain on paper towels.
To make the soup, heat the oil in a dutch oven or heavy pot over medium heat. Add the garlic and ginger and cook until fragrant, 2-3 minutes. Add the mushrooms to the pot and cook until softened, stirring frequently.
Add the chicken stock, bamboo shoots, soy sauce, rice vinegar, fish sauce, and chili garlic sauce to the pot, and stir to combine. Bring the soup to a simmer, and simmer 10-15 minutes.
In a small bowl, stir together the corn starch and water until the corn starch is dissolved and no lumps remain. Pour the corn starch mixture into the soup and return the soup to a simmer. Simmer 3-4 minutes, until slightly thickened.
Stir in the tofu, and cook until heated through. Pour the beaten egg slowly into the soup, stirring it in with a fork to break it into ribbons. Cook 1-2 minutes, until the egg is set. Remove the soup from the heat and stir in the green onion and sesame oil.
Serve hot with crispy wonton strips sprinkled on top.