Preheat the oven to 190C/Fan 170C/400F/Gas Mark 5. Scrub the carrots clean and trim away the tops, place them in a large roasting tin. Add the oil, garlic, the lemon juice, a third of the tarragon and plenty of salt and ground black pepper, then toss well to mix.
Mix the butter with the lemon zest, remaining tarragon and plenty of ground black pepper. Divide the butter into four, then use your fingers to push each quarter under the chicken skin. Lay the chicken breasts skin side up on top of the carrots. Bake for 25-30mins, until the carrots are lightly charred and the chicken golden and tender. Serve with green vegetables and new potatoes.