Soak mushrooms in a bowl with 1 1/2 cups warm water until pliable, 20 to 25 minutes. Remove mushrooms, saving liquid. Snip out stems with scissors and discard. Slice mushrooms diagonally 1 in. wide, then diamond cut.
Meanwhile, lay chicken flat on a work surface and slap with the side of a Chinese chef's knife to flatten to 1/4 in. Cut into lengthwise strips about 1/2 in. wide, then diamond cut. "I use 1 portion of meat for 3 or 4 portions of vegetables," Yan notes.
In a bowl, combine chicken with cornstarch, 1 tbsp. soy sauce, and 1/2 tsp. white pepper. Set aside.
Bring broth, mushroom soaking liquid, and ginger to a boil in a large covered pot over high heat. Reduce heat to medium-high and add carrots and mushrooms. "Don't throw in everything together or it will be overcooked. In Chinese cooking, you want everything to have crunch."
Cook, covered, for 5 minutes, then add onion, celery, and zucchini. Cover, return to a low boil, and cook until onion is tender-crisp, 2 to 3 minutes.
Add chicken, tomato, cilantro stems, remaining 1/4 tsp. white pepper and 2 tsp. soy sauce, the salt, and oil to taste. Cook, covered, until chicken is cooked through, 3 minutes more.
Using chopsticks or a fork, whisk eggs in a bowl until whites are completely blended but not so much that they foam. "If they foam, they will float on top and be messy," Yan cautions.
Turn off heat under soup. Stirring it very slowly with a ladle, gradually pour in eggs. Garnish with cilantro sprigs.
*To peel tomato, cut into wedges and slide a sharp knife between peel and flesh.