Whisk cornstarch with cold water to make a smooth slurry; set aside.
Warm oil in a large pot over medium-high heat. Add chicken and sear until just opaque, 2–3 minutes. Transfer to a plate.
Reduce heat to medium. Add garlic, ginger, and mushrooms. Cook, stirring, until fragrant and mushrooms soften, about 2 minutes.
Pour in chicken broth. Stir in soy sauce, rice vinegar, chili paste, white pepper, sugar, and salt. Bring to a lively simmer.
Add bamboo shoots and tofu. Return the chicken to the pot. Simmer gently until chicken is cooked through, 4–5 minutes.
Stir the slurry, then drizzle it into the soup while stirring. Simmer until slightly thickened, 1–2 minutes.
Create egg ribbons: swirl the soup and slowly stream in the beaten eggs, stirring gently to form wisps.
Finish with sesame oil and scallions. Taste and adjust vinegar, heat, and salt as you like. Serve hot.