In a blender combine avocado, stock and lemon juice and process for several minutes to a smooth puree; season to taste with salt and Tabasco pepper sauce. Refrigerate, covered, for at least 1 hour until well chilled.
In a nonreactive bowl combine lime juice, chile, tomato, cilantro and chives. Slowly whisk in oil; season to taste with salt and Tabasco pepper sauce. Stir fish into marinade, cover and refrigerate for 60 minutes. Remove from refrigerator and let sit 15 minutes at room temperature.
To serve, ladle soup into a large shallow soup bowl. Remove fish from marinade with a slotted spoon and mound in center of plate. Garnish with cilantro or dill sprigs and serve immediately.