In a large saucepan combine water, potatoes, green onion, chicken bouillon granules, tarragon, garlic, and pepper. Bring to boiling. Reduce heat and simmer, covered, for 20 to 25 minutes or until potatoes are very tender. Cool.
Drain the potatoes, reserving the liquid. Mash potatoes. Gradually stir in the reserved liquid. Stir in the buttermilk. Transfer mixture to a bowl; cover and chill for at least 4 hours or up to 24 hours.
To serve, ladle chilled soup into bowls. If desired, garnish each serving with additional tarragon or savory. Makes 6 to 8 side-dish servings.