1. Preheat the oven to 400 degrees F. Cut the top 1/4-inch from the garlic head, then drizzle with oil, and sprinkle with a pinch of salt. Wrap the garlic in foil, then place in the oven and roast until golden, about 40 minutes. Let cool, then peel and smash the cloves into a paste.
2. Mix half the chopped tarragon with the garlic, season with salt and pepper, then place the paste between the flesh and skin of the chicken. Slice the skin on the thighs, then coat with the paste. Sprinkle the chicken cavity with the remaining chopped tarragon, salt, and pepper. Squeeze the orange over the chicken and put the orange in the cavity. Loosely tie the legs together, drizzle with 1 tablespoon oil, and sprinkle all over with salt and pepper.
3. Cut the squash into 1/2-inch slices. Toss the squash and mushrooms in a roasting pan with the remaining 3 tablespoons oil, salt, and pepper. Place the chicken over the vegetables. Roast the chicken until a thermometer registers 160 degrees F in the thigh, not touching the bone, 50 to 60 minutes. Let stand 10 minutes before carving. Garnish with a tarragon sprig.
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