Blend first 7 ingredients in processor until coarse puree forms. With machine running, gradually add oil through feed tube; blend well. Transfer mixture to bowl; whisk in cream. Add parsley, tarragon, cilantro, basil, and shallot; whisk to combine. Season dressing to taste with salt and pepper. Cover and chill at least 3 hours . Let stand at room temperature 20 minutes and rewhisk before serving. DO AHEAD Can be made 1 day ahead. Keep chilled.
