Put the potatoes in a saucepan of lightly salted boiling water. Bring back to the boil, then reduce the heat and simmer for 7 minutes. Add the carrots and simmer for a further 5 minutes. Drain and leave until cool enough to handle.
Meanwhile, place the chicken breasts in a shallow dish. Mix together the oil, lemon juice and 1 tablespoon of the chopped fresh tarragon (or 1 teaspoon dried) with a little salt and pepper. Drizzle this over the chicken and turn to coat. Cover and set aside to marinate in the refrigerator while preparing and cooking the vegetables. Preheat the oven to 190 degrees C.
Peel and thinly slice the potatoes. Thinly slice the carrots. Whisk the sour cream into the stock, then stir in the garlic, spring onions and half the remaining fresh tarragon or remaining 1 teaspoon of dried tarragon.
Roughly layer the potatoes and carrots alternately in a lightly greased, shallow ovenproof dish, spooning a little of the sour cream mixture over each layer. Finish with potato slices covered with the last of the sour cream mixture. Cover the dish with foil and bake for 35–40 minutes.
Arrange the chicken on top of the potatoes and re-cover with foil. Bake for 25–30 minutes until the chicken is cooked through (the juices will run clear when the chicken is pierced with a skewer). Scatter over the remaining tarragon.