Heat oven to 400°F. Pat the chicken dry inside and out with paper towels. Remove giblet package, if present. Being careful not to tear it, use your fingers to gently separate the skin from the breast meat and place 4 sprigs of the tarragon under the skin on both sides. Add remaining tarragon to the chicken cavity. Rub olive oil all over the chicken and sprinkle with kosher salt and pepper. Place chicken in a roasting pan or heatproof baking dish.
Roast, uncovered, for 35 minutes. Add fennel to pan or dish; sprinkle lightly with salt and toss to coat. If chicken is browning to fast, cover loosely with foil. Roast another 25 to 35 minutes or until an instant-read thermometer registers 170°F. in the breast or 180°F. in the thigh and fennel is tender, stirring fennel once more during cooking.
Remove pan or dish from oven and let stand 10 minutes.