In a blender, puree the avocado with the buttermilk, parsley, mayonnaise, sour cream, chives, tarragon, anchovy and 1 1/2 teaspoons of the Champagne vinegar. Scrape the dressing into a bowl and season with salt and pepper. Add the chopped chicken breast and toss to coat.
In another bowl, whisk the olive oil with the remaining 2 teaspoons of Champagne vinegar. Stir in the diced cucumbers and tomato and season with salt and pepper.
Mound the baby arugula on plates. Pile the chicken salad on the arugula. Spoon the cucumber and tomato salad alongside and serve right away.
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