Put the chicken in a large saucepan, season with a handful of sea salt and cover with water. Bring to the boil over a high heat, then skim the foam that rises to the top, using a ladle. Reduce the heat so that the chicken is only simmering, then add the tarragon, bouquet garni, celery, carrots, leeks, turnips and cabbage. Push a clove into each of the onion halves and add to the saucepan. Cook the chicken for 1¼ hours until the meat is very tender and falling off the bone.
Meanwhile, make the sauce gribiche. Fill a small saucepan with water and bring to the boil. Add a teaspoon of vinegar to the water, as this will make the eggs easier to shell after cooking. Place 1 egg in a ladle, then slowly and carefully slide the egg into the water so that you don’t break the shell. Repeat for the remaining 3 eggs. Cook for 8 to 9 minutes until hard-boiled, drain and place the eggs under running cold water. When cool enough to handle, shell the eggs and cut in half.
Using a pestle and mortar or a food processor, mash the hard-boiled egg yolks until they form a smooth paste. Mix in the mustard and season with salt and pepper. While still mixing, add the vinegar, then slowly add the oil until the consistency resembles mayonnaise. Fold the gherkins, capers and herbs into the mixture and adjust the salt and pepper, if necessary. Finely chop the cooked egg whites and add them to the sauce. If you like, this sauce can be made the night before and chilled in the fridge.
Now it is time to enjoy. Serve the broth as a first course with a piece of bread – it is really good – then remove the chicken from the pan and cut it into pieces. Serve with the vegetables and Sauce Gribiche.