Peel the onion and carrot, trim the celery, rinse and coarsely chop. Put the vegetables, parsley and garlic in a pot with about 2 quarts of salted water, bring to a boil and lower the heat, cover and simmer for about 20 minutes. In the meantime, flame the chicken, rinse under running water and add to the pot.
Cook for about three quarters of an hour or until the meat is cooked and tends to detach from the bones. Remove the chicken, place in a soup bowl, wrap in cling film and let cool. Scrape the salt from the anchovies, rinse, dry on paper towels and debone. Trim the celery heart, remove any tough filaments, rinse, dry and slice. When the chicken is cool remove the bones and skin and cut it into fairly even slices.
Put the mustard in a bowl, dilute with the lemon juice and add the olive oil, salt and pepper, briskly stirring the mixture with a fork. In a larger bowl combine the sliced chicken, anchovies, celery and vinaigrette. Mix well, cover with cling film and keep in a cool place for at least two hours to marinate. Before serving, wash and dry the lettuce leaves and cut into thin slices. Spread over a platter and top with the chicken salad. Cut the tomato into thin slices, halve them and arrange around the edges of the platter.