In a large, heavy pot, melt the butter. Saute the onions until they are soft. Add the garlic, and saute for another minute or so. Add the fennel, parsley, and stock, and bring the mixture to a boil. Reduce the heat, and simmer the mixture about 20 minutes, until the fennel is quite tender. Puree the mixture in batches in a blender or food processor. Stir in the lemon juice, salt, white pepper, and tarragon. Chill the soup thoroughly. Stir the half-and-half into the soup, and adjust the seasonings. Serve the soup garnished with fennel fronds.