Preheat oven to 400ºF.
Place the potatoes, fennel, carrots in a roasting pan in an even layer, drizzle with EVOO, season with salt and pepper.
In a bowl combine the soft butter, tarragon, garlic and a little salt and a generous amount of pepper. Clean bird and pat dry. With your fingers separate the chicken breast skin from the meat, working with about 3/4 of the total amount of the butter mixture, maneuver half of it under the skin of each side of the breast. Don't make your self nutty trying to spread it all the way down the breast, the butter will melt once in the oven and do that work for you, and that's the point.