Season the chicken with salt and pepper. Melt the butter over medium heat in a saute pan just large enough to hold the chicken. Add the chicken, skin side down, and cook for about eight minutes over medium heat. If the butter in the pan starts to brown, lower the heat slightly. Turn the chicken and light saute in the same way for five minutes more.
Remove the chicken from the pan and place it in a bowl. If the butter in the saute pan has burned, pour it out and replace it with two tablespoons fresh butter. Add the leeks and cook them for 10 minutes over medium heat. Stir the leeks every couple of minutes. They should be soft but not brown.
Put the chicken back in the pan with the leeks. Add the apples, apple juice, vinegar and broth. Bring the liquids to a slow simmer and with a ladle skim off any fat or froth that floats to the top. Cover the pot and simmer the chicken very gently for 10 to 15 minutes, until it is completely cooked.
Remove the chicken and let it cool. Remove and discard the skin and pull the meat away from the bones. Cut the meat into half inch chunks.
Use a ladle to skim off any fat that has formed on the surface of the liquid in a saute pan. Add the Calvados and the heavy cream. Bring to a simmer and season to taste with salt and pepper. Heat the chunks of chicken in the soup for two minutes and serve immediately in hot bowls.
Suggestions and variations: Try adding a cup of sliced mushrooms along with the apples. The mushroom flavor goes well with the other ingredients, and the mushrooms make the soup more substantial. This soup is also delicious with curry, which can be added by gently cooking a tablespoon of curry powder in a tablespoon of butter and stirring it into the soup shortly before serving.