Remove the legs, wings and neck from the bird (keep these pieces for the hen and onion juices, right), leaving just the two breasts on the frame. Put the frame with the olive oil in a vacuum bag and seal. Cook in a water bath for 42 minutes at 64°C (147°F). Leave to cool, then chill.
To build a smoker, take a deep oven tray and build a small fire in it with the hay. Smother the fire by putting a flat tray on top of the deep tray, locking in all the smoke. Lift off the flat tray and put a wire rack with the chicken frame on the smothered fire. Replace the flat tray to lock in the smoke. Leave for about 20 minutes, then lift off and rebuild the fire. Repeat the smoking twice more.
Put the smoked chicken in the fridge to chill and then cook, skin-side-down, until it has a good colour.
While the chicken is cooking, melt the butter. Turn and baste the chicken in hot butter several times until golden and crisp. Cut the breasts off the frame, then slice and dress with the vinegar.
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