Make the dressing: whisk together olive oil, tarragon vinegar (including the extra 2 teaspoons), chopped tarragon, lemon juice, and Dijon mustard in a small bowl; season with salt and pepper to taste.
Toss the chicken: place the diced cooked chicken in a medium bowl and mix in 1/4 cup of the dressing to coat; set aside.
Cook the orzo: bring a large pot of salted water to a boil and cook the orzo until just tender but still firm to the bite (follow package time, about 8–10 minutes); drain in a colander and rinse under cold water to cool, then drain well.
Combine orzo and dressing: transfer the cooled orzo to a large bowl, add the remaining dressing, and toss to coat evenly.
Add the salad ingredients: fold in the dressed chicken, halved cherry tomatoes, drained marinated artichoke hearts, chopped Kalamata olives, dried currants, and drained capers.
Season and serve: taste and adjust seasoning with salt and pepper as needed, then serve.