Heat oil in heavy large pot over medium-high heat. Add chicken and onions and cook until brown, stirring frequently, about 15 minutes. Add 12 cups water, celery, parsley and bay leaves. Bring to boil, skimming surface. Reduce heat and simmer gently until reduced to 8 cups, about 5 hours. Strain into bowl. Cover and refrigerate until fat solidifies on top. (Can be prepared 2 days ahead.)
Remove fat form soup and reserve fat for matzo balls.
For Matzo Balls:
Place shiitake mushrooms in small bowl. Pour 2 cups hot water over. Let soak until softened, about 30 minutes.
Melt 1/3 cup chicken fat and cool. Combine melted chicken fat, 1/4 cup shiitake soaking liquid (reserve remainder), eggs, 2 tablespoons chives, 1 1/2 tablespoons tarragon, 1 1/2 teaspoons salt and 1/4 teaspoon pepper in medium bowl and beat to blend. Mix in matzo meal. Cover and refrigerate 3 hours. (Can be prepared 1 day ahead. Cover mushrooms in soaking liquid and refrigerate.)
Measure 3 1/2 quarts water into large pot. Salt generously and bring to boil. With dampened hands, form cold matzo meal mixture into 1-inch balls and add to boiling water. Cover and boil until matzo balls are cooked through and tender, about 40 minutes. (To test for doneness, remove 1 matzo ball and cut open.) Transfer matzo balls to plate, using slotted spoon. (Can be prepared 1 day ahead. Cover tightly and refrigerate.)
Drain mushrooms, reserving liquid. Thinly slice mushrooms, discarding stems. Combine remaining mushroom soaking liquid, mushrooms, chicken soup and 1 teaspoon fresh tarragon in heavy large saucepan and bring to simmer. Season to taste with salt and pepper. Add matzo balls and simmer until heated through. ladle soup into bowls. Garnish with chives and serve.