Scrape burdock root with the back of a knife to remove all skin. Shave root diagonally into narrow, thin shavings, about 1 inch long. Place shavings in a mixture of 1 cup of cold water and he vinegar. Soak 3-4 minutes; rinse under cold running water and drain.
Cut potato lengthwise into quarters; cut quarters crosswise into thin slices. Place in remaining cup of cold water; soak five minutes. Drain.
Cut daikon and carrot lengthwise into quarters; cut quarters crosswise into thin slices. Cut green onions crosswise into 1/2 inch lengths. Remove and discard mushroom stems. Cut caps into halves; cut halves crosswise into thin slices.
Place tofu in strainer; using fork, break tofu into pieces no larger than 1/2 inch. Rinse and drain.
Heat the oils in a 3-quart saucepan over high heat 1 minute; add chicken and saute, stirring constantly, until chicken is cooked, about 3 minutes.
Add burdock, potato, daikon, and carrot to chicken. Saute, stirring constantly, 2 to 3 minutes.
Add dashi, soy sauce, and salt to chicken mixture. Heat to boiling, then reduce heat to medium-high. Boil gently until vegetables are tender, about 5 minutes, skimming any foam, if needed.
Stir mushrooms and tofu into soup; cook 2 minutes. Stir green onions into soup and remove from heat.
Ladle soup into 4 soup bowls and serve immediately. Pass seven spice powder or sansho at table.