Put the chicken in a 7 to 8-quart heavy pot with the onion, garlic, and bay leaf. Pour in enough water to just cover the chicken and bring to a boil. Skim off any scum that rises to the surface. Reduce the heat and gently simmer the chicken, covered, about 20 minutes.
Meanwhile, clean the leeks: Split them lengthwise to within 1 1/2 inch of the root end, leaving the root end intact. Submerge the leeks in a large bowl of water, let them soak to release any dirt or sand, for several minutes. Lift the leeks out of the water and discard the water.
Add the leeks, potatoes, carrots, and parsnips to the chicken and continue to simmer, partially covered, until tender and the juices of the chicken run clear when a thigh is pierced with a skewer, about 25 minutes more.
Transfer chicken to a cutting board and let rest, loosely covered with foil, about 10 minutes.
Carve the chicken and arrange it on a large platter surrounded by the vegetables. Sprinkle the chicken with the coarse salt. Season the broth with salt and pepper and serve it on the side with the cornichons and cheese.
Recipe adapted from Gourmet Magazine